singular Perturbation Problems in Ship Hydrodynamics 



The normal velocity component In the plane of the vortex is: 



2 , / v2ii/2 _, , r- r 2 J. / J. \2il/2, 



V^'.o.^) =^(.^[- ^-^^^F^] -^ [-^^^^-^^^^^ 



A lifting line can be described in a similar way if we allow 

 the dipole density to vary with the spanwise coordinate, z. For 

 simplicity, let us assume that \x{z) = \i{-z) , and that |i{s) = 0. The 

 potential for a lifting line is: 



^{x,y,z) = X^f d^M)\ . '^\ ^ ^-77f (2-34) 





de 



s' Jo [(x-e)^ +y^ + (z-;f] 



3/2 



(2-35) 



and the normal velocity component is: 



Note that this reduces to the result for the single horseshoe vortex 

 if a) we set fJL'(z) = 6(z+s) - 6(z-s)*, and b) we integrate over a span 

 from -s-P to s+P, where P is a very small positive number. This 

 may lend some credibility to the procedure frequently advocated by 

 aerodynamicists in wing problems, viz. , when integrating by parts 

 in the spanwise direction, extend the range of integration slightly 

 beyond the wing tips so that quantities which become infinite at the 

 tips do not yield infinite contributions that cannot be integrated. 

 (This is terrible mathematics, but apparently the physics is sound, 

 since the results seem to be correct.) 



Finally, we can use the above procedures to derive the cor- 

 responding expressions for a lifting surface. The important quantity 

 is the nornnal velocity component, given by: 



^(x.o., = -^£ ^^o^¥^b ^ ^^-KT'^^n 



(2-37) 



6(z) is the usual Dirac delta function. 



695 



