2 

 Since n = 0(e), the above relation results <J> n = 0(e ). The oscillatory 



part of the fluid motion is composed of the incident wave potential (J) and 



w 



the oscillatory disturbance potential <j), . The former is independent of £ 

 and can be regarded as a given function. In the case of regular waves 

 propagating in the direction making an angle a with the x axis, the 

 velocity potential of the incident wave is expressed by 



<j) = h/g/K exp[Kz-iK(x cos a+y sin a)+ia)t] (97) 



w 



where h = wave amplitude 



K = wave number 2tt/A 



0) = circular frequency of encounter 

 The absolute frequency of the wave is 



co = w - U K cos a (98) 



There is a relation between the wave number and the frequency as follows: 

 K = u^/g = (oo-UK cos a) 2 /g (99) 



The order of magnitude of U and (i) or K may not be unity, so that we need to 

 include these quantities in the argument of the order. It can be assumed 

 that the amplitude of the ship's oscillation is of the same order as that 

 of the wave amplitude. The frequency of the oscillation is, however, not 

 the frequency of the wave 0) n but the frequency of encounter U). The wave 

 amplitude is of the order of S/K so that the velocity of the oscillatory 



motion of the ship has the order 03 6 K . The fluid velocity of the 



-1/2 

 incident wave has, on the other hand, the order 6 K , so that they are 



not necessarily the same order. We have to keep these facts in mind in 



examining the order of magnitude of the oscillatory part of the velocity 



potential. Taking the first order terms with respect to 6, we obtain 



41 



