singular Perturbation Problems in Ship Hydrodynamics 



that the following are appropriate simplifications: 



0^ gH(x) + ^0^$x' on y = 0; (5-18) 



The second condition is a Neumann condition; the right-hand side 

 is known, and the condition is prescribed on a known, fixed surface. 

 In fact, (5-19) is satisfied by the real part of: 



'A 



CD 



ds p(s) 



where 



z = X + iy. 



p(x) = t1q(x)^q^(x,0). (5-20) 



This follows from the Plemelj formula. (See, e.g. , Muskhelishvili 

 [1953] .) The function p(x) can be interpreted in terms of the fluid 

 velocity which is needed to correct the flow field because of the 

 error incurred by taking the free surface at y = 'nQ(x) while using 

 the potential function <^Q(x,y) to prescribe the velocity field. This 

 is the same correction which was discussed above in connection 

 with (5-8). Now we may observe that, smce t|o = 0(U ) and 

 ^Q(x,y) = 0(U), it follows that p(x) = 0(U^. Thus also: 



|V^|=0(U^ as U-*0. (5-Zi) 



This is certainly a much stronger conclusion than (5-8)! 



The integral expression given above is not the solution of the 

 $ problem, even in the first approximation, since it does not satisfy 

 the body boundary condition. However, since the existence of $ 

 arises from a defect of ^q in meeting the free-surface conditions, 

 it is difficult to imagine that the above estimate of the order of mag- 

 nitude of ^ is not correct. 



Numerical procedures could readily be worked out for 

 solving problems of the above type. In fact, all that is needed is 

 one algorithm which handles the problem of a given distribution of 

 the normal velocity component on a surface in the presence of a 

 plane rigid wall. The integral part of the solution given above would 



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