singular Perturbation Problems in Ship Hydrodynamios 



I. INTRODUCTION 



This paper is a survey of a group of ship hydrodynamics 

 problems that have certain solution methods in common. 



The problems are all formulated as perturbation problems , 

 that is , the phenomena under study involve small disturbances from 

 a basic state that can be described adequately without any special 

 difficulties. The nmethods of solution make explicit use of the fact 

 that the disturbances of the basic state are small. Mathematically, 

 this Is formalized by the introduction of one or more small param- 

 eters which serve as measures of the smallness of various quantities. 

 The solutions obtained will generally be more nearly valid for small 

 values of the parameter(s). 



However, the problems will also be characterized by the 

 fact that they are Ill-posed In the limit as the small parameter(s) 

 approaches zero. Thus, we call them singular perturbation prob- 

 lems . Special techniques are needed for treating such problems , 

 and we have two which are especially valuable: 



1) The Method of Matched Asymptotic Expansions, and 



2) The Method of Multiple-Scale Expansions. 



The first has a well- developed literature, and It has been made 

 particularly accessible to engineers by Van Dyke [ 1964] . The 

 second, which has a longer history. Is perhaps less well-known, 

 but we now have a textbook treatment of It too, thanks to Cole [ 1968] . 

 Because of the availability of such books, my treatment of the 

 methods In general will be extremely terse. 



The necessity for treating ship hydrodynamics problems as 

 perturbation problems arises nnost often In the Incredible difficulty 

 of handling the boundary condition which must be satisfied at the free 

 surface. Even after neglecting viscosity, surface tension, com- 

 pressibility, the motion of the air above, and a host of lesser 

 matters, one can still make little progress toward solving free- 

 surface problems unless one assumes that disturbances are smeill 

 -- In some sense. Historically, It has comnaonly been assumed 

 that the boundary conditions may be linearized; In fact, this has so 

 commonly been assumed that many writers hardly mention the fact, 

 let alone try to justify It. 



The two methods emphasized In this paper can also be applied 

 to problems Involving an Infinite fluid. In fact, neither method was 

 applied specifically to free-surface problems until quite recent 

 times. Section 2 of this paper Is devoted to several Infinite -fluid 

 problems. My justification, quite frankly. Is almost entirely on 

 didactic grounds. The methods can be made much clearer In these 

 simpler problems , and so I include them here, although In some 



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