Submerged Two-Dimensional Bodies 



From this extended research, we may be able to draw conclusions of great 

 importance to the surface-ship case. 



ADDENDUM 



The author would like to mention briefly some additional work completed 

 after the preparation of this paper. This additional work is included here in re- 

 sponse to the discussion following this paper. 



It is shown in this paper that at the free-surface and in the far-field the 

 complex potential has the two following second-order terms: 



2w 



the second-order free-surface term, 



ewg , the second-order body correction term. 



The author only included one of these second-order contributions, namely the 

 free-surface term e^wp . The only justification for this was that Tuck (4) has 

 shown that for a circular cylinder this is the most important higher-order con- 

 tribution to the wave resistance. On the other hand, Giesing and Smith (5) have 

 developed a method which neglects the second-order free-surface term e^Wj, 

 but includes the second-order term from the body boundary condition ewg . 



The author has applied the method by Giesing and Smith and combined it 

 with the method presented in this paper, resulting in an entirely consistent 

 second-order theory. The wave resistance computed from the consistent 

 second-order potential is shown in Fig. 19. Comparing Fig. 13 and Fig. 19 it is 

 seen that both second-order terms are important and that neither should be neg- 

 glected. It is especially interesting to note that at lower speeds (u/N/ib" < 0.65), 

 the main higher order contribution to the wave resistance comes from the free- 

 surface term; however, at higher speeds (u/\/ib" > 0.75) it is seen that the body- 

 correction term gives the essential contribution. This is an interesting result, 

 considering that in the case of the circular cylinder Tuck showed that the free- 

 surface contribution was the most important for the entire speed range. 



The reason that the body -condition term is so important for the wing-shaped 

 body treated here is that in addition to the singularities introduced in closing 

 the body a circulation term must also be introduced such that the Kutta condition 

 is satisfied at the trailing edge. It can be shown that it is mainly this circula- 

 tion term which gives rise to the large higher order effect at higher speeds 

 (u/^iF > 0.75). 



ACKNOWLEDGMENTS 



To Professor Finn C. Michelsen and Professor Chia-Shun Yih I express my 

 deep gratitude for their unending inspiration, encouragement, and guidance in 



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