Ogilvie 



asymptotic expansions does not offer the possibility that one may be 

 able to obtain higher-order approximations. 



In fact, the problem of a thin body in an infinite fluid is not 

 a genuine singular perturbation problem (although it may contain 

 some sub-problems that are singular, such as the flow around the 

 leading edge of an airfoil). However, I believe that the problem of 

 a thin ship is singular; I shall discuss this in Section 4. There has 

 been a considerable amount of misunderstanding as to what consti- 

 tutes the near field and what constitutes the far field in the thin-ship 

 wave- resistance problem, and the rectification of such misunder- 

 standing requires a careful statement of the problem. 



It is conceivable that this interpretation of the thin- ship 

 problem may be useful in formulating a rational mathematical 

 idealization of the maneuvering- ship problem. 



For convenience, I separate the thin-body problem into two 

 parts: a) the symmetrical-body problem, and b) the problem of a 

 body of zero thickness. To treat an arbitrary thin body, with both 

 thickness and camber, one should certainly consider both aspects 

 at once. It is not really difficult to do this, and indeed the problemi 

 of an unsymmetrical body of zero thickness actually involves thick- 

 ness effects (at higher orders of magnitude than in the symmetrical- 

 body problem)^, I have kept the problems separate here only for 

 clarity in discussing certain phenomena that occur. 



2.11, Symmetrical Body (Thickness Effects) . Let the body 

 be defined by the equation: 



(2-1) 



for (x,0,z) not in H, 



where H is the part of the y = plane which is inside the body. 

 (It is the centerplane if the body is a ship.) The "thinness" of the 

 body is expressed by writing: 



h(x,z;e) = eH(x,z), (2-2) 



where 6 is a small parameter and H(x,z) is independent of €. 

 The body is immersed in an infinite fluid which is streaming past it 

 with a speed U in the positive x direction. The flow, in the 

 absence of the body, can be described by the velocity potential: Ux. 



It will sometimes be convenient to say that the body is defined 

 by the equation: y = ± h(x,z;e), innplying that the function h(x,z;e) 

 is identically zero if (x,0,z) is not in H , Also, note that we shall 



680 



