4. FUNDAMENTAL INTEGRAL IDENTITIES 

 In this section, basic integral identities for the velocity potential are 

 obtained by applying a classical Green identity to the potential (p = (})(x) and the 

 previously defined Green function G = G(?,x). The Green identity is 



j ((j)V^G-GV^(f))dv = I" ((j)9G/9z-G8())/9z)dxdy 



+ J (G3(l)/9n-({)9G/9n)da + | (cj)9G/9n-G9cj)/9n)da (4.1) 

 h h 



00 



where d^ is the finite domain bounded by the ship hull surface h, the mean sea plane 

 z = 0, and some arbitrary, but sufficiently large, exterior surface h surrounding 

 the ship surface h, as is shown in Figure 1; furthermore, a' is the portion of the 

 plane z = between the intersection curves c and c of the plane z = with the 



oo ^ 



ship surface h and the exterior surface h , respectively. On the surfaces h and h 



■^ oo' r J OQ 



we have 9())/9n = V(})'n and 9G/9n = VG*n where n is the unit outward normal vector to 

 h or h^, as is shown in Figure 1. Finally, dv and da represent the differential 

 elements of volume and area at the integration point x of the domain d and the sur- 

 faces h or h , respectively, and dxdy is the differential element of area of the 

 mean sea surface o^ . 



Let the integrand ())9G/9z - G9(j)/9z of the sea-surface integral in Equation (4.1) 

 be expressed in the form (p[d -(f+iF9 +ie)^]G - G[9 -(f-iF9 +ie) ]()) + 

 2iF(f+ie)9(G(l))/9x + F 9 (G9(J)/9x-(f)9G/9x)/9x. Furthermore, we may use the relation 



12 



