INTRODUCTION 



A body in waves is subject to not only first order zero mean forces proportional 

 to wave amplitude but also to second order forces which are proportional to the square 

 of the amplitude. In the case of a body free to move, the motions will result in 

 hydrodynamic reaction forces which have both first and second order components. 



The second order forces are quite small compared to the first order forces, and 

 in seakeeping work are generally neglected. In some circumstances, however, they can 

 cause significant effects, and must be accounted for. If a force or moment component 

 is not opposed by a corresponding hydrostatic or other force or moment, over a long 

 time span large motions can result, as in surge, sway, and yaw of a ship at zero speed 

 (Maruo, I96O) or heave for a body submerged but near the surface (Newman and Lee, 1971)' 

 Added resistance in waves is also a problem for moving ships (Strom-Tej sen et al . , 1973] 

 Semi -Submers i bles , which have a small water-plane area relative to the volume, are 

 frequently subject to a tilt in waves which is believed to be caused by second order 

 forces (Numata et al., 1976). Moored vessels, with low natural frequencies of the 

 vessel -moor ing line system, can be excited at resonance by the low-frequency components 

 of the second order force (see for example, Hsu and Blenkarn, 1970). 



In the last decade, major advances in the understanding of and ability to predict 

 these second order wave induced forces have been made. In this paper, developments 

 in the specific area of slowly-varying drift forces will be reviewed. 



EXACT REPRESENTATION OF THE SECOND ORDER FORCE 



If the seaway is assumed to have a discrete spectrum, the wave height can be 

 represented as 



C(t) = Re Z A e'^^m*^ (1) 



m 



where A is the complex amplitude (with random phase) of the wave component of frequency 



ca . The force on a body, through second order, can be represented as 

 m 



F(t) = F^^^ (t) + F^^^ (t) (2) 



where the first order force is 



F^^^t) = Re E A H^^^o) ) e'V (3) 



m m 



and the second order force is 



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