singular Perturbation Problems in Ship Hydrodynamics 



The most Interesting feature of this inner expansion of the 

 two-term outer expansion is that the wave effects are all contained 

 in g(x) --a function of just x. In the infinite -fluid problem, all 

 3-D effects in the near field were included (in the first approxinmatlon) 

 in the single function of x, f|(x). We now have a genereilization of 

 this for the free-surface problem. 



In the near field, it is easy to show that the first ternn in the 

 asymptotic expansion of the potential is again just the uniform- stream 

 potential, Ux, The next term, 4>|, must satisfy the Laplace 

 equation in two dimensions (in the cross -plane) and the same body 

 boundary condition as before, (2-63): 



a$. ur (x,e) 



[1 Mro^Ao)^]' 



r=rQ(x,e). (3-10) 



As in the infinite -fluid problem, this conditions suggests that 



^, = 0(e2) , 



since tq = 0(e) and 8/8N=0(€''). 



Now consider the free- surface conditions. In the Bernoulli 

 equation, note the orders of magnitude: 



gt, + U$,^ +^ (*f^ + ^fy + ^f^) + ... = on z = ;(x,y). 



0(C) O(c^) 0(€^ 0{€^) 0{€^) 



2 

 The term containing ^i^ can be dropped, but the others containing 



^1 are all the same order of magnitude, and we have no reason to 



suppose that the t, term is higher order. In the kinematic condition, 



note the orders of magnitude: 



U^K +\;x +^ly^y - *l, + ... =0 on z=;(x,y). 

 o(u 0(;€2) 0(U 0(c) 



Clearly, we can drop the ternn containing ^ij^ , but no others. 



Now we must relate the order of magnitude of t, with the 

 order of magnitude of ^| . From the kinemiatic condition, one nnight 

 suppose that t, = 0(€). However, the dynannic condition then implies 

 that t, ~ 0, which means only that C, is higher order than we 

 assumed. In fact, the only assumption which is consistent with both 

 conditions is that: 



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