*(?) 



> = i)(0 - UK-A) for K < 



<t>(K)/2 



ind + a-h-c 



ind. + O . -h-c 

 1 1 



exactly on h + c 



(4.10a) 



> (4,10b) 



(4.10c) 



where d. and a. represent the domain and the portion of the plane z = 0, respec- 

 tively, strictly inside the ship surface h, as is shown in Figure 1. 



The value of the constant C on the left side of Equation (4.7) is discontinuous 

 across the ship hull surface h; C being equal to 1 outside h and to inside, as is 

 explicitly indicated in Equations (4.10a, b, and c) . This discontinuity in the value 

 of C evidently is accompanied by a corresponding discontinuity on the right side of 

 Equation (4.7). Specifically, the latter discontinuity stems from the dipole- 

 distribution integrals j, ct)3G/8nda and J (fi8G/8xt d£ in the potential L(^;())) 



defined by Equation (4.9). An identity valid for any point ^ — outside, inside, or 

 exactly on the ship surface h — can be obtained by eliminating the discontinuity in 

 the value of C in Equation (4.7). This can be done by adding the term C (j)^ on both 

 sides of Equation (4,7), with C. given by 



C. = f V^Gdv - f [3^-(f+iF8^+ie)^]Gdxdy 

 d. a. 



(4.11) 



Use of the divergence theorem 



f V^Gdv = r 3G/3zdxdy + 8G/3nda 



17 



