Ogilvie 



we are recilly saying is this: From very far away, the disturbance 

 appears as if it could have been generated by a sheet of sources, 

 but close-up we allow for the possibility that this observation from 

 afar may be somewhat inaccurate. In fact, there is no analytic 

 continuation presumed in the present method. 



One can show by the use of Green's theorem that the far- 

 field picture is valid even if the analytic continuation is not possible, 

 A particularly appealing (to me) version of such a proof has been 

 provided by Maruo [ 1967] for the much more complicated problem of 

 a heaving, pitching slender ship moving with finite forward speed on 

 the surface of the ocean. 



I suppose that the uniformity of the thin-body solution is the 

 result of the fact that a well-posed potential problem can be stated 

 by giving a Neumann boundary condition over a surface. The situ- 

 ation will be quite different when we consider slender -body theory: 

 in the far field, it would be necessary to give boundary conditions 

 on a line, and that does not lead to a well-posed potential problem 

 in three dimensions. Similarly, we may expect trouble at the con- 

 fluence of two boundary conditions , and this indeed occurs when we 

 try to treat a ship problem by the method discussed above. The 

 free-surface conditions cannot be satisfied, and the difficulty can 

 be traced back to the behavior of the far-field potential near the 

 intersection of the centerplane and the undisturbed free surface, 



2.12, Unsymmetrical Body (Lifting Surface) . For the sake 

 of simplicity, let the body have zero thickness. Then it can be 

 represented as follows: 



y = g(x,z;e) = €G(x,z) for (x,0,z) in H » (2-23) 



where H is now the projection of the body onto the y= plane. 

 Again, there is a uniform incident flow in the positive x direction. 



The analysis is quite similar to the symmetrical-body case, 

 at least in the near field, and so most ot the details will be omitted 

 here. In the near field, let there be an expansion: 



N 



<^(x,y,z;c)~ ) ^n(x,Y,z;e), 



n=0 



just as in (2-16). The first term is, again, $o(x»Y,z;c) = Ux. The 

 terms again satisfy the transformed Laplace equation, (2-17): 



[L] $, + $2 + *3 + *4 + 



'yy yy yy ^yy 



^ [^Ixx +*lzz +^2xx +^2zz + '••] ; 



690 



