298 



Vg- • V({.rr n dS = [V- (V4)n • n 



+ V(f.j7 (V3- • n)] dS 



[V3- {V<f„- • t) 



(A-13) becomes 



•(2) 



(On V(t.s + 0g V(f,^)dS 



3<}) 



+ P 



V<t.c 



n dS 



3z 



(A-15) 



+ V<j>n- (V3- • n) - V^ 



s "fn 



n]dS 



(A-9) 



The last two terms in the integral over Sg combine 

 to yield 



- (n X V) X (fn Vg- 



(A-10) 



The first term on the righthand side of (A- 10) 

 vanishes since (f)^ = on Sq . The second term also 

 vanishes, since by Stokes' theorem 



{n X V) X ^- Vc 



dx X (Jjjj V3 =0 (A-11) 



The first term has the same structure as the steady 

 flow Lagally force derived by Lin (1974) for a 

 linearized source sheet representation of a slender 

 strut piercing the free surface. The second term 

 arises from the intersection of the hull with the 

 free surface in unsteady flow. 



In the low Froude number approximation, 3cfig/3z 

 = on z = (rigid wall representation of the 

 free surface), and G(x,x') -+ 0. In this case (A-2) 

 and (A-3) yield 



fS 4^ 



Og(x') 



Ix-x'. I 3 

 ' 1 ' 



dS 



and 



where the contour Cq is taken as the hull waterline. 

 Consequently, using (A-6) , (A-9), and (A-10) , the 

 expression for the force becomes 



4Tr 



a„(x') 



x-x' 



x-x'. 



dS 



(2) 



[V3 {V<t>n • n) + V^p- (V3- • n) 



+ V(!>p + V(t.r 



(A-16) 



Og G^ n] dS + p 



V3- (V(|. - • n)dS {A-12) 



The contribution from the free stream, iu(in Vg) , 

 vanishes since 



iU(V(j)n • n)dS = iU 



V^^^ dV = 



for X on S, and where the integrals are to be 

 interpreted in the principal value sense. Inserting 

 these expressions into (A-15) and performing the 

 integrations, the equation for the force reduces to 



F '^> = - P 



ag(V(j)p + V(j)p ) dS 



n In 



S+Sq V 



Also noting that Vijij^ • n = - a^ and V3 • n = - 03, 

 (A-11) reduces to 



3(}i 



+ P 



^S 3l 



dS 



(A-17) 



(2) 



[On V<t>3 + Og V(j)n + 03 On n] dS 



+ P 



'*s" ''*'n ■ n <iS 



(A-13) 



which is the result given as Eq. (42) in the text. 

 The reduction in the first term reflects the fact 

 that there is no net force arising from the mutual 

 interaction of the body sources. 



or, upon defining 



v*3+ + v^; 



APPENDIX B 

 REDUCTION OF THE ANALYSIS OF PROPELLER INDUCED 

 VERTICAL SURFACE FORCE TO AN INFINITE FLUID 

 PROBLEM 



The linearized unsteady pressure at a point, x, 

 on the ship hull surface is given by (8) as 



V*n 



V(() + + Vtj) - 

 n n 



(A-14) 



p(x,t) 



3t 



(x,t) + V„ (X) ■ 



|(x,t) 



(B-1) 



