Ogilvie 



restraining effect of gravity (to this order of magnitude). 



This condition points to a fundamentally different kind of 

 solution from that of the previous problems. If we continue the 

 function <^| analytically into the upper half- space, it must be odd 

 with respect to the surface z = 0. Thus 4>| cannot represent a 

 line of sources. The least singular solution represents a line of 

 dipoles, oriented vertically. Assuming that <)>| will consist only 

 of such dipoles , we can write it: 



*,(x.y,z)=lflif d6,(e) ['^ I'l ,,, ]. (3-16) 



*^0 ^ L (x-9) + r J -■ 



where y = r cos 9 and z = r sin 0, The two-term outer expansion 

 and the two-term inner expansion of the two-term outer expansion 

 are, respectively: 



4)(x,y,z) ~ Ux + <^|(x,y,z) 



^ ^^ ^ 2 sin 9 t defi.(e). (3-17) 



"■ ^0 



I am now assuming that the bow of the ship is located at x = 0; then, 

 in matching to the near -field solution, we can show that the dipole 

 density must be zero upstream of the ship bow. This expansion is 

 unaffected by the downstream dipoles. 



In the near field, we assume the usual expansion: 

 N 

 <^(x,y,z) ~ Ux + y $n(X'y»2) for fixed (x,y/e,z/€). 



The term, ^, satisfies the 2-D Laplace equation: 



^1 + 4>, =0. 



yy 11 



The body boundary condition suggests that $, = 0(e ), just as it did 

 before in Eq. (3-10). 



From the dynamic free-surface condition, 



= gC + U$i^ +-^ (^ly + ^1^) on z = ;(x,y), 



we see that t, = 0(e) (since g = 0(e) ). This causes a new problem. 

 We would like, as usual, to change this condition at z = C(x,y) to a 



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