singular Perturbation Problems in Ship Eydrodynamios 



smaller with each iteration. Such a procedure is discussed, for 

 example, by Wehausen and Laitone [ I960] , who point out the use- 

 fulness of Kochin functions in such procedures. However, there is 

 often a question about the precise nature of such expansions. In the 

 first approximation, for example, the effects of the free -surface 

 are likely to drop off exponentially with distance froin the surface. 

 This makes it inappropriate to treat depth of submergence as a large 

 parameter in the usual manner, because exponentially small orders 

 of magnitude are either trivial or exceedingly difficult to handle, 

 I do not believe that anyone has yet shown how to treat this problem 

 systematically. 



II. INFINITE- FLUID PROBLEMS 



It is mainly the presence of the free surface in our problems 

 that forces us to seek ever more sophisticated methods of approxi- 

 mation. However, the nature of the approximations can often be 

 appreciated more easily by applying those methods to infinite -fluid 

 problems. In this section, I discuss a number of problems that are 

 geometrically similar to the ship problems that are my real concern. 

 In some cases, it must be realized that the methods used here are 

 not necessarily the best methods for the infinite -fluid problems. 

 However, without the complications which accompany the presence 

 of the free surface, one can better understand the significance of 

 the coordinate distortions, the repeated re-ordering of series, and 

 the matching of expansions. 



The reader who feels comfortable with matched asymptotic 

 expansions is invited to skip this chapter. 



2.1. Thin Body 



A "thin body" has one dimension which is characteristically 

 much smaller than the other dimensions. In aerodynamics, the 

 common example is the "thin wing," and, in ship hydrodynannics , 

 one frequently treats a ship as if it were thin. In such problems, 

 the incident flow is usually assumed to approach the body approxi- 

 mately edge-on, and so the thinness assumption allows one to 

 linearize the flow problem. 



In this section, thin-body problems are treated by the method 

 of matched asynnptotic expansions. This is not the way thin-body 

 problems are normally attacked, and, in fact, I do not recall ever 

 having heard of such a treatment. At the outset, I must point out 

 that there are good reasons why this has been the case. If the body 

 is symmetrical about a plane parallel to the direction of the incident 

 flow, one does not need inner and outer expansions for solving the 

 problem. And if the body lacks such symmetry, the lowest-order 

 problem cannot be solved analytically, and so the method of matched 



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