Linearized Potential Flow Theory for ACVs in a Seaway 



This is on the outer hull surface S. . In the case of the other surfa- 

 ce S the expansion would be 



a G r 



a G r 



n + bs-ah, ($'$■') 



+ 5 



r^h ^G Sh ^G 



1 r r 



?7=— b_ 



B\(*'n 



as' a* 



2 



-N G 



^7,2 



»? = -b_ 



af a r J 



+ ou 2 ) 



o G 



+ 6 



■a^ ac r ^ ac r 



~aT aT + aF aT 



TJr-b- 



- h x (rr) 



r, 2 J 



Similar expansions with h instead of h. will apply to the two inner 

 hull surfaces S + and S, 



The normal derivarive has thus been reduced to the derivative in the 

 lateral direction across the longitudinal planes with the addition of 

 0(5) terms which are only required when we have to evaluate the 

 0(5/3) potential. 



Let us now consider the singular part of G , namely l/r . 

 As in the case of G 



4- <-r> =T f- <"T> +s 



dn r drj r 



r dh > . ah ^ ■ 



_1_°_ /JL\ + L J_ (Jl\ 



scarr' ar ar l r ; 



If the point (x, y, z) at which the potential is to be determined is far 

 removed from ( £ , V , f) which is confined to the hull surfaces for 

 the purposes of integration, there is no difficulty, for l/r is then 

 regular and may be considered as part of G_ in the above expansion. 

 We shall, however, be actually interested in the case when (x, y, z) 

 also lies on the hull surfaces for the determination of the potential 

 and thereby the pressure thereon which causes the forces and moment. 

 A closer examination of the normal derivative is therefore necessary 

 in this case in view of the possible singularity at (x = £ , y = V , 

 z = O . 



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