Vassilopoulos and Mandel 



In support of your contention that our approach is incorrect, you cite the 

 work of Professor Fay among others. May I quote Professor Fay's discussion 

 of Korvin-Kroukov sky's 1955 SNA paper? In that paper the forces due to body 

 motions are developed through the following equations: 



The potential 



^bm = (V^-2-^^)r COS a (37) 



and since 



the pressure 



p 



-^ ^ W tan /3 and -^ = V , 

 dt ^ dt 



-^ = cos a(^r +5V2 tan/S- zr - zV tan/S-eVr- ^5r -e^V tan/S) . (38) 



The vertical force increment per unit length 



I 



dF 

 dx 



/ 2 



= 2r I p cos a da 



becomes 



f.- K^^v)^W.|rV^tan,).-(p- 



- (p|-rV tan/3jz - p| r^yj^ - L^ r' A'S - (p^r^V tan ^U . (39) 



Professor Fay said: "If e is positive when measured clockwise, z is posi- 

 tive in the downward direction, and V positive for motion in the positive 

 x-direction then Eq. (37) is correctly stated. However, d^/dt should equal -v, 

 and terms (1) and (5) in (39) do not cancel but add. This term is the most im- 

 portant coupling term in the equations of motions and exists even for a sym- 

 metrical vessel." He also commented, with regard to the terms in dr/dt, that, 

 since the method is a linear approximation, "the carrying of terms of higher 

 order in subsequent equations does not seem justified." 



In the 1957 SNA paper by Korvin-Krovikovsky and myself, we corrected the 

 sign of V, and reinstated the velocity-dependent terms, which had been omitted 

 in the 1955 paper on the assumption that these terms in the potential theory de- 

 velopment merely implied damping and could be replaced by damping terms de- 

 termined on the basis of energy dissipation by waves, as a quid pro quo . A 

 study of Haskind (1946) and Havelock (1955) confirmed what Fay had said in his 

 discussion about the coupling terms. The Korvin-Kroukovsky approach now 

 has values for the coefficients e(aj^) and E(oj^) , as shown in your Table 2, 

 which contain the identical dynamic coupling terms derived by Havelock for a 

 long half-immersed spheroid and by Haskind for a thin "Michell" ship. 



358 



