pressure on the hull surface, such as the added mass and damping. Numerical 

 computations are not so simple and may require much computer time, if the 

 forward velocity is taken into account, but they are possible at any rate. 

 There has been some attempt of calculating the added mass and damping of 

 a ship making forced heaving and pitching oscillations during forward 

 motion in still water. However, it has failed to obtain any useful result 

 because the numerical results show very unrealistic features, in spite of 

 a rational appearance of the formulation. This fact will be discussed in 

 another section. 



Various Cases of the Order of Magnitude 

 of Frequency Parameter and Froude 

 Number 



In the preceding section, we have assumed that the frequency parameter 

 of oscillations and Froude number are both of the order of unity, and it 

 was expected that the theory was valid without any restriction in magni- 

 tudes of frequency and forward speed. However, the results were quite 

 disappointing. It was noted, on the other hand, that the different choice 

 in the assumption of the order of magnitude of frequency parameter or 

 Froude number might result in different formulation. The discussion of the 

 order of magnitude in the boundary condition may derive different solutions 

 for different assumptions. Here we will consider cases that the Froude 

 number is not so large, or the frequency of oscillations is not so small. 



Now let us begin with the case of low Froude number. The Froude 



number of conventional merchantile vessels is not much greater than 0.3, 



2 -1 -1 



so that the speed parameter U /gl = Yn is of the order of 10 , which may 



be regarded as a small parameter. Therefore, let us assume that U//g£ = 



1/2 

 0(e ). The order of magnitude of the radiation potential is 6e but the 



1/2 

 effect of the forward speed appears in the term of the order 6 e . The 



lowest order term in the free surface condition has the order of <5 and the 



next order is 6e. If we take up to the order of 6e, the free surface 



condition in the near field becomes 



2 2 



9 (J). 3<J). 3 <J>_ 



—7T +g^ + U ~^r " ° at z = (128) 



~,_2 3z w ~ 2 

 3t 3z 



49 



