and L'(C;(J)) is the linear transform of <t) defined as 



L^d;*) = I ((})-(f)^)8G/8nda - 2i(f+ie)F ( G((})-cf)^)t dZ 

 h c ^ 



+ F^ ( [((})-(() )8G/9x-G(t 8(j)/9£-n t 3(})/9d)]t d£ (4.15) 



J*^ xzy y 



in which we have (}) = (j)(x) and (j)^ = (})(^) as was defined previously. In obtaining 

 Equation (4.13), the relation C + C. SI was used. This relation can be obtained by 

 using Equations (3.3a and b) and (3.4a and b) in Equations (4.5) and (4.11), which 

 yield 



+ C^ = r V^Gdv - j [9^-(f+iF3^+ie)^]Gdxdy 

 z<0 z=0 



Identity (4.13) is valid for any point E,, whether outside, inside, or exactly on the 



ship surface h. This identity thus is essentially equivalent to the set of the three 



classical identities (4.10a, b, and c), which are exclusively valid for 5 outside, 



inside, and on the hull surface h, respectively. 



Identities (4.10a, b, and c) and (4.13) correspond to the case of an open hull 



surface piercing the sea surface. For a closed, fully submerged surface h, the 



waterplane integral w(^) defined by Equation (4.14) and the integrals around the 



mean waterline c in Equation (4.8) for the potential ip(C) and in Equations (4.9) 



and (4.15) for the potentials L(5;(j)) and L (^;(J)) are evidently not present. Two 



other important particular cases of identities (4.10a, b, and c) and (4.13) are 



obtained in the limiting cases when the Froude number F vanishes, corresponding to 



9 

 wave radiation and diffraction by a body with zero mean forward speed, and when 



the frequency parameter f vanishes, corresponding to steady flow about a ship 



advancing in calm water. 



19 



