singular Perturbation Problems in Ship Hydrodynamics 



non- uniqueness are especially great. The history of the study of 

 free- surface problems provides numerous examples of invalid solu- 

 tions being published by authors who were not sufficiently careful on 

 this score. We have learned to be careful about imposing a radia- 

 tion condition when necessary, although newcomers to the field are 

 still occasionally trapped.'"' Questions about stability of our solu- 

 tions are not so well appreciated, but of course solution stability is 

 just one aspect of solution uniqueness. A particularly startling 

 example has been pointed out in recent years by Benjamin and Feir 

 [ 1967] : Ordinary sinusoidal waves in deep water are unstable. This 

 has now been demonstrated both theoretically and experimentally. 

 It comes as no great surprise to those experimenters who had tried 

 to generate high-purity sinusoidal waves for ship-motions experi- 

 ments , but it was certainly quite a surprise to the theorists, who 

 apparently did not suspect any such phenomenon before its discovery 

 by Benjamin and Feir. 



Since we shall be considering snnall -perturbation problems, 

 we may expect the solutions to appear in the form of series expres- 

 sions (not necessarily power series I). Often, we are content to 

 obtain one term in such a series. Practically never do we face the 

 question of whether the series converges. In fact, we usually just 

 hope that the series has some validity, at least in an asymptotic 

 sense. 



The question will arise from time to time, "How small must 

 the small parameter be in order that a one- (or two- or three- or n-) 

 term expansion give valid predictions?" In ship-hydrodynamics 

 problems, it is quite safe to assert that the only answer to such a 

 question must be based on experimental evidence. In fact, even in 

 simple problems , the knowledge of a few terms is not likely to help 

 much with this question. For example, suppose that one tries to 

 solve the simple differential equation: y"(x) + y(x) =0, by means of 

 a series of odd powers of x. How does one know that a two -term 

 approximation is accurate to within one per cent even if x is as 

 large as unity? One might compute the third term, of course, and 

 compare it with the second term, hoping to guess what the effect 

 of further terms would be. K it were too difficult to compute that 

 third term, one could only hope that the solution had some validity, 

 and perhaps one would try to find some experimental evidence on 

 which to hazard a guess about validity. So it is in our ship- hydro - 



Within the last few years, a leading German journal published an 

 article on wave resistance in water of finite depth, in which it was 

 concluded that a body had identically zero resistance if it were 

 symmetrical fore and aft. The author was, I believe, primarily 

 a numerical analyst, not familiar with the pitfalls of free- surface 

 problems. He did not impose a numerical condition equivalent 

 to a radiation condition. (This is one reference that I intentionally 

 omit. ) 



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