Singular Perturbation Problems in Ship Hydrodynamics 



The body is at rest in our reference frame. 



The rigid-wall solution satisfies all conditions of the free- 

 surface problem except the dynamic condition on the free surface. 

 The latter could be used to define the free -surface shape. Thus, if 

 the free-surface disturbance is expressed by 



Ti(x) ~ r)Q(x) + r),(x) + . .. , 



the dynamic free- surface boundary condition says that: 



Ti(x) ~ Tio(x)= Ti^^^" H(x,0)]. (5-7) 



Of course, the kinematic condition is now violated, but an additional 

 velocity field which is 0(U ) can correct that. And so it appears 

 plausible that: 



<Hx,y) - %{x,y) = o(U). (5-8) 



One point should be noticed from this conclusion. The limit 

 process "U "* 0" implies that Froude number goes to zero. Nothing 

 has been said about the length sccile used in defining Froude number, 

 but it does not matter so long as all dimensions are fixed. The 

 submergence and the body dimensions may be quite comparable, 

 for example. Thus, we are not considering t/b as small, in the 

 sense that Salvesen did. However, bjjth t and b are supposed to 

 be large conapared with the length U /g; we imply this if we state 

 that all dimensions must be fixed as U ""*• . 



It would be wrong to take <^x,y) as the potential for the flow 

 around the body in an infinite fluid (without either free surface or a 

 rigid- wall substitute). The body can be quite near to the free 

 surface in Salvesen's sense, and so the effect of its image cannot be 

 neglected. Furthermore, at least part of the effect of the image is 

 0(U), even if the body is very far away from the free surface, and 

 such an effect must be included in the first term of the approximation 

 which is supposed to be valid as U — ^ 0. 



The next problem is to find [ <f)(x,y) - 4>o(x,y)] . We consider 

 two possible approaches in the following subsections. 



5.41. A Sequence of Neunnann Problems. As above, let there 

 be a velocity potential , (^(x,y), which provides the solution of the 

 exact problem: 



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