water and a train of regular waves comes in the direction perpendicular to 

 the axis of the body. The incident wave is partly transmitted beyond the 

 body but partly reflected, generating reflected waves which propagate in 

 the direction opposite to the propagation of the incident wave. At a 

 great distance from the body, there exist regular sinusoidal waves. On the 

 weather side, there are incident waves and reflected waves, while on the 

 lee side, there are transmitted waves. If we assume vertical planes on 

 both sides at a great distance from the body, the momentum flux across 

 these planes can be evaluated simply by the expression for regular waves. 

 If we write the amplitude of the incident wave by h, that of reflected 

 wave by h^, and that of transmitted wave by h , there is a relation for the 

 energy conservation law 



h 2 = h 2 , + h 2 (236) 



Now we apply the momentum principle to the fluid between the vertical 

 surfaces and take the time average for one period of the wave. The result 

 shows the steady horizontal force D in the direction of the propagation of 

 the incident wave experienced by the floating body, the amount of which is 

 given by a simple relation 



D=TPgh! (237) 



Since the first order forces are periodic, the steady force is the second 

 order force. Nevertheless, it is calculated by the first order solution 

 of the diffraction problem. A similar analysis can be applied to the 

 three-dimensional problem even when the forward speed is present. 



Instead of the actual case in which the ship penetrates into waves 

 with a velocity U, we consider a ship floating on a uniform stream with a 

 train of regular waves. A constant horizontal force is assumed in such a 

 way that keeps the average position of the center of gravity of the ship 



86 



