singular Perturbation Problems in Ship Hydrodynamics 



modified condition at z = 0. But this is not possible. For example, 

 the term ^j would be transformed: 



$,^(x,y,^(x,y)) = $,^(x,y,0) + C(x,y)^,^^ (x,y ,0) + ... . 

 0{€^) 0(c2) 0(e) 0(€) 



Every term, in fact, will be the same order of magnitude, and so this 

 ordinary kind of expansion fails. We must continue to apply the con- 

 dition on the actual (unknown) location of the free surface. 



The kinematic free -surface condition is also nonlinear and 

 must be satisfied on the unknown location of the free surface: 



t^ U^x + <^ly^y- ^Iz on z = C{x,y). 



Each term here is 0(e) , and so none can be ignored. 



We are left in the rather uncomfortable position of having to 

 solve a nonlinear problem just to obtain a first approximation to the 

 near-field potential function. However, that nonlinear problem is a 

 two-dimensional problem, which is not an insignificant advantage, 

 and, as we shall see, it is possible in principle to predict the loca- 

 tion of the free surface, thus avoiding the necessity of searching for 

 it. 



We do not have a condition to apply at infinity in the $| 

 (near-field) problem. It is not so straightforward in this case to 

 predict the form of the solution as r -♦ oo , but Ogilvie [ 1967] 

 showed that: 



$,(x;y,z)= AlLillli. [i + 0(l/r)] 



oo , 



where A|| is a constant to be determined. There is no source-like 

 behavior. This might have been expected, of course, since the inner 

 expansion of the outer expansion, (3-17), showed the characteristics 

 of a two-dimensional dipole. An intermediate expansion can be 

 used to show that these statements are correct. 



A numerical procedure for solving this problem may be the 

 following: Suppose that at some x we know the value of $| on the 

 free surface, z = C,(x,y), and that we also know £,(x,y) at that x. 



If we expand: ^ ~ 2j^n» "we could apply the condition on z = ^\ , 

 then apply the usual kind of transformation, as above, so that 

 conditions on higher-order terms would be applied on a priori 

 known surfaces. 



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