Ogilvie 



W = i c^2a.2b*. = -p I (D:, <t),„ dS 



/ ^t^J-in 



Thus the diagonal damping coefficients can be calculated from the velocity po- 

 tential at infinity for this mode of oscillation. 



Next, let just two a. 's, say a^ and a^^ be non-zero. We can choose the 

 relative phases so that £3-65 = ^/2 or 0. In these two cases, then, from the 

 respective potentials at infinity we can calculate respectively 



w = f -. 



[-'^Mlb-Mb.) - (^ab-^ba)] • (21) 



W -- |-'^a% (b:b+b;,) . (22) 



From the latter, we obtain the sum of any symmetric pair of coupling damping 

 coefficients, but it is not generally possible to find them separately. 



If the ship has zero forward speed, then c . ^ is just the hydrostatic coupling 

 coefficient, which is easily calculated from the ship lines. In this case we can 

 use such a calculation, together with (21), to find ^*^ - /x*^, and from the equa- 

 tions (previously proved) relating added mass and damping coefficients we can 

 in principle calculate the difference between the two damping coefficients: 



^] [^ab^'^') - ^ba^^')] 



da' 



(It may be recalled that h^^ = t>*j^(oo) = for zero forward speed, and also 

 '^ab = ^\h^^^ - /^ba-) ^ thi^ special case, we now have both the sum and the 

 difference of b*^(aO and b^^Ca) , from which they can be individually calculated. 



These formulas are useful generally only when we have found the appropri- 

 ate velocity potentials for the oscillatory ship motions. The problem of finding 

 these potentials will be taken up in the next two chapters. In the meantime, it 

 may again be pointed out that all of the results of the present chapter require 

 the knowledge of the potentials only at a great distance from the ship. Further- 

 more, there are no problems here in deciding whether the pressure must be 

 evaluated on the actual hull position or the mean hull position. This is a great 

 simplification in carrying out computations, but, as has been seen, there are 

 several gaps in the results. In particular, the coupling damping coefficients 

 cannot be found in the case of non-zero forward speed, and therefore the added 

 mass coupling coefficients cannot be determined either. These gaps would not 

 exist if we could calculate the c.^'s, but doing this is a rather formidable 

 undertaking; it must be recalled that these coefficients are not just the hydro- 

 static coupling coefficients, but, rather, they depend strongly on hydrodynamics 

 and they involve the complicated local flow around the ship. 



48 



