Ogilvie 



terminate it. The one s implification which is admissible here is to 

 evaluate f(z) and its derivatives on z =Z,, where t, = Z| + o(e).* 



I have not worked out any more terms in any of these expan- 

 sions, but I suppose that the next term in this near-field expansion 

 will be much more interesting. In the far field, it is well-known 

 that the third term in the expansion of the potential function will in- 

 clude the effects of what appears to be a pressure distribution over 

 the free surface. It was shown by Wehausen [ 1963] that at the inter- 

 section of the undisturbed free surface and the hull surface the solu- 

 tion is singular, and he represented the singular part by a line 

 integral taken along this line of intersection. From the point of 

 view of the method of matched asymptotic expansions, it should be 

 possible to represent the far- field effects of that line integral in 

 terms of an equivalent line of singularities on the x axis. The 

 strength of the singularities would be determined, as usual, by 

 matching the solution to the near-field expansion. At this stage, 

 thin-ship theory will have become a singular perturbation problem. 



V. STEADY MOTION IN TWO DIMENSIONS (2-D) 



Sometimes we study two-dimensional problems with the intent 

 of incorporating the solutions into approximate three-dimensional 

 solutions, as in the treatment of high-aspect-ratio wings and in 

 slender-body theory. And sometimes we investigate two-dimensional 

 problems simply because the corresponding three-dimensional prob- 

 lems are too difficult. 



The problems discussed in this section are in the second 

 group. It is not likely that any of these problems and their solutions 

 will have practical application before several more years have 

 passed, even in the context of strip theories. Here are some of the 

 most fundamental difficulties related to the presence of the free 

 surface. 



The first two subsections concern a 2-D body which pierces 

 the free surface. Such a problem is intrinsically nonlinear. We 

 might try to formulate the problem as a perturbation problem, in 

 this case involving a perturbation of a uniform stream. However, 

 there must be a stagnation point somewhere on the body, and at that 

 point the perturbation velocity is equal in magnitude to the incident 

 stream velocity. It is not small I If the stagnation point is near the 

 free surface, the free-surface conditions cannot be linearized. We 

 must find methods which are adaptable to highly nonlinear problems. 



Such a method is the classical hodograph method, used since 



The same difficulty arises also in Sections 3.2 and 5.42. 



776 



