Vassilopoulos and Mandel 



It has been indicated by others, that integrating the expression on the left 

 by parts will reduce it to the right-hand expression provided the sectional area 

 curve goes to zero (continuously) at the ship ends. Since this is a requirement 

 of slender body theory, the left-hand terms can be written in the simpler form 

 of the right-hand term. Similarly, it can be shown that the two terms under the 

 Korvin Approach for Coefficient "B", and indicated by the dotted block in the at- 

 tached figure, reduce to the one term, indicated by the dotted block under "New 

 Approach." Some ships, such as those with transom sterns, need not have sec- 

 tional area curves which are zero at the stern end, hence the possibility of an 

 error in Korvin Approach for this hull shape. On the other hand the Korvin Ap- 

 proach gives a distribution of the effect along the length, which is desirable 

 when bending moments are being considered. 



There are only two additional coefficients in the tabulations which take dif- 

 ferent forms under "New Approach" and "Korvin Approach" — these are coeffi- 

 cients C and E. For coefficient E, (or -M^), the Korvin approach has the addi- 

 tional term 



-I 



dx 



X dx 



as compared to the "New Approach" and this term reduces to -u^ji^x) dx or 

 u^Z.. Since the Korvin approach is a slender-body theory involvii^ some pseudo 

 three-dimensional effects, it will be shown below that this additional term can 

 result from three-dimensional considerations. As introduced by Korvin, coeffi- 

 cient b (or - Z^) is expressed by jN(x) dx which is purely a frequency depend- 

 ent effect (surface wave effect) since in potential theory, for a deeply submerged 

 body b (or z^), would be zero— i.e., no lift force with angle of attack in the ab- 

 sence of circulation. Hence, in the attached table the term zero has been added 

 to indicate the addition of a three-dimensional potential solution. If we include 

 in the pure strip approach or "New Approach" column, the other three- 

 dimensional potential solutions in the appropriate terms, the following terms 

 are added to the expressions in the "New Approach" column: 



where -x. is the added mass for longitudinal acceleration. 



u 



These additional terms appear in the attached table as encircled by a dotted 

 line. The z. terms are equivalent to the terms enclosed by dotted rectangles 

 under the Korvin approach. However, we now find terms in X^ in the "New Ap- 

 proach" brought about by the rough three-dimensional correction based on the 

 results of potential theory calculation. Since X. can be estimated for a given 



362 



