identical with the shape of the corresponding body generated by the same 

 singularities in an unbounded fluid. 



It is well known that in the latter case one can construct the 

 contour of a body of revolution for any given singularity distribution along 

 the axis; auxiliary tables for this work are available, 2 ' 10 especially for 

 cases in which the distribution is given by polynomials. Plat noses— as 

 discussed by Weinstein 16 - will not be dealt with in the present report, 

 although it is possible that such forms are advantageous from a point of 

 view of wave resistance at high Proude numbers. When dealing with "normal" 

 shapes, the important approximation developed by Weinig 7 and Munk 8 holds; 

 i.e., for very elongated bodies the sectional-area curve of the generating 

 body A(x) is affine to the doublet distribution n(x) . This approximation 

 will be used throughout the present report although its limitations should 

 not be forgotten. 



Some explanation -if not definition -must be given as to. the concept 

 of a "normal" shape of a doublet -distribution or a sectional -area curve. It 

 means essentially a curve whose trend is similar to sectional-area curves of 

 common ocean-going ships; these curves generally are monotonic with not more 

 than one point of inflection in the fore and afterbody. 



Since for closed bodies the source-sink distribution a (x) is the 

 derivative of the doublet distribution //(x) the latter is monotonic over the 

 range of the forebody when <r(x) consists only of sources in the same range. 

 This condition (though not necessarily a required one) is sufficient to ob- 

 tain bodies such that the circle of curvature at the nose lies inside of the 

 meridian contour. 



We mention some conditions under which the affinity between the 

 doublet and the sectional-area curve becomes strained: 



a. For larger values of the elongation D/L the divergence between the 

 sectional-area curve A(x) and the doublet distribution u(x) becomes more 

 pronounced even for "normal" shapes. This divergence can be roughly de- 

 scribed. : First , in the mutual relation of the prismatic (area) coefficients 

 which are the decisive form parameters of the two curves — the one,^, de- 

 noting the prismatic or area coefficient of the distribution, and the other, 

 <f> , the corresponding one for the sectional -area curve — the following state- 

 ment holds for a wide class of normal bodies: 5 ' 15 

 for finite D/L 



<£ s > <£ d when d < 2/3 

 <f> s < <j> d when <£ d > 2/3 



