Eggers 



The above expressions can in general be evaluated in closed form for mathemat- 

 ical elementary hulls, save the contributions of the local flow ^A^ '^ to the inte- 

 grands, where however the v integration may be interchanged with closed-form 

 ^,C integration. 



Resistance Due to Additional Singularities at the 

 Undisturbed Free Surface 



Consider a strip of width d^ extending from 77= -T to 77= T at ( = with or- 

 dinate X = ^. Assume that a Fourier expansion for §(^,17) holds as 



^^'\i,V) -- Y^ K(^) cos (U,7o^)- 



(39) 



( 2) 



If Xg « ^ , this strip will contribute to 1/^2 ^Y 



d0 



(2 



T 

 ) r S free 



= ^:;;7 Gx (^,7j.0,X,Y,Z) dT7d^ 



J.J 



with 



(40) 



free 



CD 



y(^'^ 



W,7o(X-^) 



COS (U,7oY) cos (U^ro77) (41) 



(+ terms odd in r] not needed here), where 



g, = g.^ = 477K, 7o'/^'vT 

 (compare Eq. (Al) in the appendix). We then have 



(42) 



K.rnZ 



d02^^ = - 2T 2^ (S.KyM,T) e"-'«^ [cos (W^7„^) cos (W,7oX) 



+ sin (W^ro^) sin (W^7oX)] cos (U^7oY) d^ 



(43) 



K now for ^\^^ as well, only free waves are significant at X= X^, i.e., if we have 



,rnZ 



4'' = 4'^'^'^ = 2Z k'' COS (W.7oX) + Bi'^ sin (W,7oX)l e''- " cos (U,7oY) , 



V=- 00 L 



(44) 



( 2 ) ( 3 ) 



then di/^j will make up a contribution to Rj as 



662 



