Recent Progress in the Calculation of Potential Flows 



(e) 



-2.0 



^/V»-U.0902 



-1.5 



Fig. 14 (Continued), (e) Radial - y onset flow, Vq perturbation 



If our method of attack were applied to the problem of the wave resistance 

 of three-dimensional bodies, such as ships, and if a linearized free-surface 

 boundary condition were used, we would replace our simple source distribution 

 by a distribution of Havelock sources (Wehausen and Laitone, Ref. 9). The 

 source function can be written as 



q)^(x,y,z) = - - — + (p(x,y,z) 



(20) 



where 



q)(x,y,z) = lim Re 



u->0 



J -T, Jo 



,k(R+ia.)dk 



k - (v sec^f? - i/zsec 0) 



Some familiarity with this formula is assumed; therefore the terms will not be 

 defined. To evaluate this integral in a rapid and accurate manner, it is desir- 

 able to approximate the integrand with functions that can be integrated analyti- 

 cally. With that in mind, we see that the integral presents two difficulties: 

 first, the integral over the variable k develops a singularity as fi -* o; second, 

 there is the fact that the exponential is complex and does not lend itself to a 

 polynomial approximation. Both difficulties can be overcome if a contour in- 

 tegration is performed in the complex k- plane. This process leads to the fol- 

 lowing transformed equation: 



(x,y,z) = 



^ 



'(y+b) r 



Jp sec^e U^ + 



du 



d0 



(y+b) 



[vsec^(0) pcos (0- a)] (21) 



(Cont) 



351 



