Ogitvie 



Table 3-1 shows ajj and bjj for the three problems. In Cases 1 and 

 2, the results have been obtained from Sections 2.32 and 3.3, re- 

 spectively. For Case 3, the present problem, the lengthy derivation 

 will be found in Ogilvie and Tuck [ 1969] . Some points should be 

 noted: 



1. All of the terms* in Case 3 Include the corresponding 

 Case 2 terms, i.e. , the added mass and damping at forward speed 

 can be computed in terms of the added mass and damping at zero 

 speed, plus a speed- dependent component. Formally, we could also 

 say that Case 1 includes all of the Case 2 terms, with n(x) set 

 equal to zero. From this point of view, tl^ only differences among 

 the three cases are the forward- speed effects. 



2. The coupling coefficients b35 and b53 include a forward- 

 speed term ^^^33 ^^ both Case 1 and Case 3. This means, first 



of all, that there can be some damping even in the infinite -fluid 

 problem. Secondly, it means that this contribution to the damping 

 coefficients is not altered by the presence of the free surface. Note 

 that in neither case is it necessary to ignore the steady perturbation 

 of the incident stream (the x terms in (3-41), for example) in order 

 to obtain this result. 



3. The other coupling coefficients, a^g and ag^, contain 

 similar speed-dependent terms in Case 3; they arise at the same 

 point in the analysis as the terms discussed in 2 above. We could 

 arbitrarily include such terms, ±{\j/(ji)h^y in Case 1 too, without 

 causing any errors since bj^ is zero anyway in Case 1. 



4. In Case 1, there is a speed- dependent term in agg which 

 is lacking in Case 3. The reason for the lack is that such a term 



is higher order in terms of c in the ship problem, because of the 

 assumption that u = 0(c''/^). There was no need for a high-frequency 

 assumption in Case 1, and so the extra term could legitimately be 

 retained. 



5. If, In Case 3, one arbitrarily Includes the forward-speed 

 term, -(11/(0)^333, In the agg coefficient, making it identical to the 

 Case 1 coefficient, then It Is consistent to modify bgg In a similar 

 way, namely by changing It to: 



bgg = y dx x2n(x) - (U/a))^b. 



33 



The relationship between these forward-speed effects Is quite the 

 same as that discussed above In paragraphs 2 and 3. In the bgg 

 coefficient of Case 1, we could also Introduce an extra term, 

 -(U/a))^bg3, without causing any error, since b^^ Is zero anyway In 

 this case. Thus we can maintain the symmetry between Case 1 and 

 Case 3. 



766 



