Lee 



the beating frequencies of (<r, - <t^ /Z and jr, - 0-2. Although the 

 values of C2 and C^ are not shown In Fig. 10, they are found from 

 Lee [ 1968] to be about the same order of magnitude as ^g^ and 

 Tgj^. This means that the dominant contribution to the free- surface 

 wave elevation comes from the first-order term whose beating fre- 

 quency is (a, - ^^/Z, 



We can conclude that the hydrodynamic force and the outgoing 

 waves associated with the difference -frequency of two nearly equal 

 frequencies are much smaller than the corresponding first- order 

 quantities ajid are of the same order of magnitude as the other second- 

 order quantities. An examination of the figures suggest that if the 

 difference-frequency is sufficiently small an estimate of the effective 

 "steady" force can be obtained by doubling the pure steady force. 

 However, since the magnitude of the difference-frequency force is 

 always larger than the steady component this estimate may be a 

 low one. 



ACKNOWLEDGMENT 



The author would like to express his gratitude to Professors 

 T. F. Ogilvie and J. N. Newman for their encouragement and Vetluable 

 discussions during the course of this work. 



REFERENCES 



Frank, W. , "Oscillation of cylinders In or below the free surface 

 of deep fluids , " Report 2375, Naval Ship Research and 

 Development Center, 1967, 46 pp. 



Grim, O. , "Non-linear phenomena in the behavior of a ship in a 

 seaway," presented at the iZth International Towing Tank 

 Conference, Rome, Italy, 1969. 



Hasselmann, K. , "On nonlinear ship motions In Irregular waves, " 

 J. Ship Res., Vol. 10, No. 1, pp. 64-68, 1966. 



Lee, CM., "The second-order theory of heaving cylinders In a 



free surface," J. Ship Res. , Vol. 12, No. 4, pp. 313-327, 

 1968. Also published as Report No. NA-66-7, College of 

 Engineering, University of California, Berkeley, 1966, 

 151 pp. 



Llghthlll, M J. , "On waves generated In dispersive systems by 



travelling forcing effects, with applications to the dynamics 

 of rotating fluids," J. Fluid Mech. , Vol. 27, Part 4, 

 pp. 725-752, 1967. 



940 



