singular Perturbation Problems in Ship Hydrodynamias 



a rapid transition from conditions at the surface to conditions Inside 

 the bulk of the fluid. From our experience with viscous boundary- 

 layers , we should expect the occurrence of large derivatives In this 

 region and also some difficulty In satisfying boundary conditions on 

 a face of the boundary layer. 



In a viscous boundary layer, of course, the derivatives are 

 much greater In one direction than In another, and this fact allows 

 us to stretch coordinates anlsotroplcally and apply the limit pro- 

 cesses of the method of matched asymptotic expansions. In the free- 

 surface boundary layer, however, this does not appear to be a 

 possible approach. From the linear theory, we expect that there 

 will be a wave motion with wave lengths which are 0(U /g). Thus, 

 derivatives will be large In at least two directions Inside the boundary 

 layer --In the direction normal to the layer and In one direction 

 parcillel to the layer. 



When I tried to solve this problem two years ago (see Ogllvle 

 [ 1968] ), I did not apply very systematic procedures. Rather, I 

 simply assumed that the first approximation to ^ , as defined In 

 (5-14), would have certain properties , namely, 



<&(x,y) = 0(U^; ^x(x,y), ^y(x,y) = 0(U^; 



also, the surface deflection function would be given by (5-15), with: 



H(x) = 0(u''); H'(x) = 0(uV 



The order of magnitude of $ was chosen just so that the velocity 

 components would be 0(U^, and I assumed that differentiation 

 changes a quantity by i/U In order of magnitude. The arguments 

 leading up to (5-21) contributed heavily to the conjecture about veloc- 

 ity components, and the i/U^ effect of differentiation was chosen 

 just because the free-surface characteristic length Is U /g. It Is 

 Important to note that the rigid- wall potential, <|>q. Is still part of 

 the solution, and these statements about orders of magnitude and 

 differentiation do not apply to It. In fact, I assume that ^q Is com- 

 pletely known, and so It Is not necessary to conjecture about the 

 effects of differentiation. 



In terms of the general approach of the multiple- scale ex- 

 pansion method, I have assumed that an approximation to the solution 

 can be represented as the sum of two functions. The first depends 

 only on the length scale appropriate to the body geometry. The 

 second function depends primarily on lengths measured on a sceile 

 appropriate to UVg> t>ut It also depends on the first function and 

 thus on lengths typical of the body. However, It seems to be 

 possible to keep clear when differentiations are being carried out 

 with respect to each of the length scales. 



799 



