2 2 



2z g 2 g 2x g 2xx 



(y<^^-\}^} • n = (15) 



Equations (10), (11), and (12) comprise the potential function due to the steady 

 forward motion and Equations (13), (14), and (15) comprise the potential functions 

 due to oscillation with steady forward motion. 



VELOCITY POTENTIAL OF STEADY FORWARD MOTION 

 The solutions of Equations (10), (11), and (12) lead to Michell's integral in 

 the thin-strip wave resistance problem. In the context of slender-body theory, 



however, this three-dimensional problem becomes a two-dimensional problem which has 



4-6 

 been discussed in depth by Tuck. The solution of two-dimensional potential is 



given by 



log r + f log r, - Ux (16) 



where a = — S'; the strength of source 



S = — TTr : area of immersed cross section 

 2 o 



—9 9 2 



r = (y-n) + (z-C) 



rj = (y-n)^ + (z+O^ 



Here, (y,z) is the point where the potential is solved and (ri,^) is the source point. 



UNSTEADY POTETIAL DUE TO OSCILLATION 

 The solution of Equations (13), (14), and (15) is a three-dimensional problem. 

 This can be solved by distribution of the three-dimensional source strength over 

 the surface of the ship and by numerical solution of the integral equation. How- 

 ever, this method requires considerable computer time in numerical integration. To 



3 

 avoid such a lengthy process, the slender-body theory has been introduced by Newman 



