Ogilvie 



cases the infinite-fluid problems can be treated adequately by more 

 elementary methods. 



Most of the material in this paper has appeared in print 

 elsewhere. My intentipn has been to present a coherent account of 

 the treatment of singular perturbation problems in ship hydrody 

 namics, and so I have reworked solutions by other people and put 

 them into a common notation and a common format. In some cases, 

 I have made conscious decisions to follow certain routes and to 

 ignore others, I am sure that I have made nnany such decisions 

 unconsciously too. I have tried to give credit where it is due, but 

 I am also sure that I have committed some sins of omission in the 

 references. I apologize to those whom I may have slighted in this 

 way. 



i. 1 . Nature of the Problenas and Their Solutions 



We never really derive the perturbation solution of the exact 

 problem; we derive, at best, an exact solution of a perturbation 

 problem. That is, we formulate an exact boundary -value problem, 

 simplify the problem, solve the simplified version, and then hope 

 that that solution is an approximation to the solution of the exact 

 problem. 



Thus , there will almost always be open questions about the 

 validity of our solutions, and these questions can only be resolved 

 through comparisons with exact solutions and experiments. We can 

 have little hope of being rigorous. In fact, it is difficult to provide 

 completely convincing arguments for doing some of the things that 

 we do; in many cases , our approach is justified by the fact that it 

 works I Much progress has been made in this field by people who 

 try approaches "to see what will happen. " 



This does not imply that we shoot in the dark. It does sug- 

 gest that we often depend more on intuition (or experience, which is 

 the same thing) than on mathematical logic in deciding how to solve 

 problems. The small disturbance assumptions by which free-surface 

 problems have traditionally been linearized must have been tried 

 first on this basis. The predictions which result from making such 

 assumptions agree fairly well with observations of nature, and so we 

 are encouraged to go on making the same assumptions in new prob- 

 lems. We may expect to be successful sometimes. 



There are also open questions about the uniqueness of solutions, 

 Engineers do not often worry about such matters, but they shoiald 

 certainly be aware of certain situations in which the dangers of 



"Exact" means only that nonlinear boundary conditions are treated 

 exactly. I neglect viscosity, surface tension, compressibility, 

 etc. , and still call the problem "exact. " 



668 



