singular Perturbation Problems in Ship Hydrodynamics 



1^1 " P[y\ ~* '^ ^^ r -♦ oo, 



that is, there is a uniform stream at infinity, moving at a right angle 

 to the actual incident uniform stream. This is the downwash velocity. 

 With this condition at infinity known, the $| problem can be solved 

 by the same method used for the $q problem, and all of the terms 

 in (2-53) are then known, except A _, 



We have all of the information available to match the three - 

 term outer expansion of the two-term inner expansion with the two- 

 term inner expansion of the three-term outer expansion. Using 

 (2-45c) and all of the terms in (2-54), we obtain the equation: 



Ux+Tiotan' X-fAoo+P.y+^^^i-^^^+^^^^ + S, log r + ^i, tan"' X 

 + A,o= Ux+lv,(z) [l --^tan' ^1 + mMJlJSMi) 



rf4^^4veU)[-^an'x]. 



The unknown quantities are: A,q, \x^ , \^, y^. This equation is satis- 

 fied only if: 



fj,,(z) = 2TrBQ,; \,(z) = 2ttAq,. 



From this matching step, we see that all quantities introduced so 

 far are now completely known. There is no source strength in the 

 second approximation ; there is a correction to the vorticity in the 

 far-field description; there is a correction to the "constant" In the 

 near-field problem; and the density of both vertical and longitudinal 

 dipoles in the far field is known. It is interesting to note that the 

 last were determined entirely from the lowest-order near-field 

 solution, that is, from ^q. When quadripoles first enter, it will be 

 found that they too are determined in strength from ^q solely. 



The next term would be much more difficult to obtain, since, 

 in the near field, it entails solving a Pols son equation In which the 



It would have been possible to eliminate the 6 log r terms In both 

 problems above by noting that the body boundary conditions eillow for 

 no net source strength. 



70 7 



