^J, 



lim 



dx e M"(x) 



where 



M(x) = V o-(x,y,z) dc (38) 



^c(x) 



This reduces the influences of the line singularity approximation of 

 ship hull singularities in two ways, (i) by the factor 1 /b , which 

 corresponds to cos in the Havelock wave resistance formula, 

 (2) by the factor 1 /k^ which is the smaller if the Froude number is 

 the smaller. Of course, the number of terms have been increased 

 in the singularities along c(0) , which is the intersection of the free 

 surface and the transom stern, and c(-i), which is the straight bow 

 stem line. 



VII. STREAMLINES 



To establish the validity of the mathematical model of the ship 

 with the transom stern, the double model [9, 10] scheme is inadequate 

 because free-surface waves play a vital role in the flow field of a 

 transom stern. 



Fortunately a slender ship model [ 11,12] gives an easy repre- 

 sentation of the wave height along the train of ship. The wave height is 



Ux,,y,) = ^(x,,y, ,0) (39) 



and 



^(x|,y,,0) = - -;j^J J (r(x,y,z') dS 



S 

 r*Tr pOO jt<j 



X Re \ sec^e \ ^ dt dG (40) 



•"Ltt "^o t - ksec'^O - i|j. sec 



where 



w = (x, - x) cos + (y, - y) sin 9 (41) 



and S is the ship surface. 



The inner double integral was treated by Havelock and was 

 represented by both a Bessel and a Struve functions, [ 13,14] We 

 will consider, for simplicity, a source distribution which becomes 



5b2 



