singular Perturbation Problems in Ship Hydrodynamics 



o-(x) = Us'(x), (3 -13b) 



just as In the infinite -fluid problem. Again, we have been able to 

 determine the complete two-term outer expansion without explicitly- 

 solving the near-field problem. This occurs because the source- 

 like behavior which dominates far away from the body (still in the 

 near-field sense) cbji be found simply in terms of the rate of change 

 of cross section, and it provides all the information needed for 

 determining the two-term far-field expansion. 



Enough information is now available to determine a first 

 approximation of the wave resistance. It can be computed in either 

 of two ways: 1) integrate the near-field pressure over the hull 

 surface, or 2) use the far-field expansion and the momentum 

 theorern. In either case, one obtains: 



'w 



D^A/ = wave resistance 

 7 ^ ^ 



^ p ^ J d<r(x) do-(e) Yo(/c|x-||), 



-00 -00 



where 



(r(x) = source density, given in (3-13b), 



K = g/u^ 



J / I do-(x) J 

 d(r(x) = ^^ dx, 



Yq(z) = Bess el function of the second kind, of order zero, 

 argument z. 



This is the slender-body wave -resistance formula which is so 

 notoriously inaccurate. At speeds for which one would hope to use 

 it, it gives values that are too high by a factor of 3 or more. 

 Generally, one could not (and should not) expect to correct such 

 errors by including higher-order terms, and so it is rather futile 

 to pursue this analysis further. 



Streamlines, Waves, Pressure Distributions . I mentioned 

 previously the apparent paradox of prescribing a rigid-wall free- 

 surface condition, then using the solution of that problem to compute 

 wave shapes, as in formula (3-12). Such a procedure really can be 

 quite rational. 



Once a velocity potential is known everywhere , it is a fairly 



737 



