Ogilvie 



condition will be appropriate and we proceed to use it without further justifica- 

 tion. However, the forward speed problem would require much more care. 



With 1 now replaced by s^, Eq. (25) is much simpler, but we still can do 

 nothing with it in its present form. We assume that 30/3n is a known quantity 

 on S^, but 4>(x) is not known, and so this is an integral equation for <^(x) on s^ .* 

 To reduce it to a simpler integral equation, Newman now introduces the slender- 

 ness parameter. This is essentially a rather tedious exercise in estimating the 

 relative sizes of various quantities, and I shall be satisfied here to state his re- 

 sult. He finds that ^(x) can be written as the sum of two terms plus an error: 



^C'^) = [^2dC?) + f(?^>] [l + 0(e log e)] . (26) 



where ^jd ^^ ^^^ solution of the two-dimensional problem in which the body 

 cross section performs its motions in the presence of a rigid wall at X3 = 0. 

 The function f(xj) is obtained explicitly: 



L/ 2 



f(xi) = I r F(^i){-H„(K|xi-^J) - Y„(K|xi-^J) + 2iJ„(K|xj-^J)} d^^ 



*^-L/ 2 



L/ 2 2I X - ^ I 3 



where 



c 

 C is the contour around the hull in the cross section at x^ , 



K = a;Vg , 



J^ = Bessel function of the first kind, 



Yq = Bessel function of the second kind, and 



Hq = Struve function. 



It is easily seen that F(Xj) is just the fluid flux through the hull surface at the 

 cross section at x^; it is zero for yaw and sway motions but not for heave, 

 pitch, and roll motions. (We avoid mention of surge motions. They can be ana- 

 lyzed by slender body theory just as well as the other kinds of motion, but the 

 results can be expected to be rather special with respect to orders of magnitude. 

 To some extent this is true for roll also.) 



*In order to obtain the integral equation on s^, we must let x approach S^^ , and 

 then the factor 1/47T changes to 1/277. 



62 



