singular Perturbation Problems in Ship Hydrodynamics 



;,y,z) ~ ) ^n(x,y,z), 



N 



^(x,y) ~ ) Zn(x,y), 

 n = l 

 We assume right away that: 



n=0 



for fixed (x,y/e,z). 



(^Q(x,y,z) = $Q(x,y,z) = Ux. 



In the far field, the ship vanishes as e — ^ , and so we take 

 the entire outside of the plane y = (below the free surface) as the 

 far field. It is easily seen that the second term in the outer expan- 

 sion must be of the form: 



<j>,(x,y,z) = - ^ J J o-,(e,;)G(x,y,z;e,0,;) de d^ , (4-1) 



where H is the portion of the centerplane of the ship below z = 0, 

 (r|(x,z) is an unknown source density, and G(x,y , z;^ ,r| , ^) is the 

 usual Green's function for a linearized problem of steady motion 

 with a free surface. It has the important property: 



G^^-^kG^ = Q, on z = 0, (4-2) 



where k = g/U . Of course, the potential ^\, also has this property: 



For later convenience, we define 



a,(x,z) = ^,(x,0,z), (4-3) 



and so a|(x,z) has the property too: 



'^"xx "^ '^'^Iz "" ^^' °^ z = 0. (4-4) 



With 4'|(x»y»z) given by (4-1), the two-term outer expansion 

 is: 



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



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