Kaplan 



system performance. The methods to be applied should recognize 

 the problem as Involving complicated two-point boundary value prob- 

 lems, or alternatively another technique that replaces the equations 

 by a set of difference equations or differential- difference equations 

 similar to the case of a beam vibration problem can be applied (see 

 [iO] for a discussion of different computational techniques for this 

 problem). The solution of the class of partial differential equations 

 given above is a specialized simulation problem that is the subject 

 of presently on-going research so no further detailed discussions can 

 be given. The development of these equations is another illustration 

 of the application of knowledge of hydrodynamics of ship motion 

 toward other related problems of engineering significsuice. 



IX. DRIFT FORCES DUE TO WAVES 



When a floating vessel is acted upon by waves it experiences 

 forces and moments that are predominantly oscillatory-like in 

 nature, with the frequency characteristics similar to that in the 

 spectrum of the oncoming wave system. These forces are also 

 linear with regard to wave amplitude. In addition there are also 

 nonlinear force contributions that arise from the presence of the 

 vessel hull modifying the Incident waves by virtue of Its function 

 as an obstruction, as well as the effect of Interaction between the 

 vessel naotlon and the Incident waves. These nonlinear forces are 

 much smaller than the linear wave forces, but nevertheless exert a 

 significant effect on certain degrees of freedom of the vessel. 



The major nonlinear drift forces of Importance to the problem 

 of maintaining a desired position In a seaway are In the longitudinal 

 and lateral directions relative to the vessel, as well as the yawing 

 moment that tends to rotate the vessel In heading. Somie theoretical 

 studies of these quantities have been made, but only for determining 

 average values In regular waves, with the work of Havelock [ 14] , 

 Maruo [ 15] , Hu and Eng [16] and Newman [17] serving as typical 

 examples. 



Havelock [14] treats only the drift force In head seas. His 

 formula Is based on the heave and pitch motions sind their relative 

 phase to the Incident wave. The theoretical approach of Hu and Eng 

 [ 16] , which follows that of Maruo [ 15] , yields expressions only for 

 the lateral drift force and draft yaw moinent In waves. (Maruo s 

 results only considered the lateral drift force,) While their results 

 are quite general, they have only been reduced to workable formulas 

 under the restrictive assumptions of a thin ship, with small draft. 

 In long waves. These results Indicate Infinite (practically unrealis- 

 tic) forces and moment as the wave length goes to zero. The maxi- 

 mum lateral drift force occurs In beam seas (varying as sln^P), 

 and the maximum moment occurs at an angle of 45° (varying as 

 sln^P), 



1068 



