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APPENDIX A 

 THE LAGALLY FORCE ON A FLOATING BODY REPRESENTED 

 BY A SURFACE DISTRIBUTION OF SOURCE SINGULARITIES 



tives of these functions on each side of the surface 

 S are related to the source strengths in the form 



vj' = V ~ + na^ (A-4) 



The force F 



(2) 



arising from the convective term 



of the linearized unsteady pressure, Eq. (35), is 

 given by 



F,(2) 



n dS 



(A-1) 



where, as before, the symbols 



and ( ) denote 



quantities inside and outside the hull surface, S. 

 We assume that the solutions for V^ and (f:-^ are 

 known in terms of distributions of source singular- 

 ities and images over the surface S as 



fn+ 



from which it follows that 



X on S 



• ^*n+ 



Vc 



(A-5) 



(A-6) 



since V^ • n = V<i>n • n = 0. 



We now apply Green ' s theorem to the functions 

 Vs" and V((ijj" in the closed volume V surrounded by 

 the surface S and Sq, where Sq is the hull water- 

 line plane, obtaining 



S+Sp 



■ ndS = 



V(V" 



*s(x) = - ^ j°3(xM 

 ^ S 



T^ + ^ + G(x,x' ) 



x-x ' I X-X ' i I 



♦ ntx) 



4ir 



On(x') 



dS 



x-x' x-x' 



(A-2) 

 dS + ((ip 



[V-- V(V(j)n-) + Vcjin'-VVg-] dV 



(A-7) 



since V x Vg = V x V^)" = in V. Using Gauss' 

 theorem and the fact that V • V^- = V • V$j^- = 

 in V, (A-7) may be written as 



n dS 



(A-3) 



in which Og (x) and Oj^(x) are the source singularity 

 strengths, xl is the image point of x, and G(x,x') 

 is the "wave potential" of a source located at x' 



and is regular in the half plane z 



< 0. 



The deriva- 



S+Sp 



[Vg" (V(|)n" • n) + Vcfn" (Vg • n) ] dS (A-8) 



S+So 

 and hence 



