Nonsinusoidal Osoillations of a Cylinder in a Free Surfaae 



TABLE 2 

 The magnitudes of hydrodynamic forces for \^ = 10b and X = 1. 1 



^^' 2pgb- 



It Is clear from Table 2 that the first-order force dominates 

 the second-order forces. For instance, if we assume € = 0.1 the 

 ratio of the first-order force to the largest second-order force is 

 2c |f2| /4e |f^| « 16. It is also clear that the magnitudes of the 

 difference-frequency force and the steady force are much smaller 

 than the first-order force, so they appear unimportant. However, 

 when such forces act upon a body which has very small restoring 

 force for a sufficiently long period of time a considerable excursion 

 from its mean position can occur. One can see from Figs. 5 through 

 7 that the | f g| is larger than the \ij\ in 1.0<X<2.0. This 

 means that for a sufficiently small vailue of the difference-frequency 

 an estimate of the maximum "steady" force acting on an oscillating 

 body should include the difference-frequency force. 



If we assume that the motion of a wave maker is described 

 by Eq, (80), the expression for the free-surface elevation, Y(x) , 

 for large x can be given in the form 



Y(x) ~ e2C, cos ""' ~ ""^ t cos {K^\x\ - p, - a^t) 



+ €^|4C2 cos {a^ - (T^)t cos (4K2|x| - ^2 " ^^z^^ 

 + 4C3 cos {o■^ - or2)t cos (2K2|x| - P, - 2cr2t) 



+ Yg(x) cos (a - <T^)t f + 0(e3) (82) 



where C,, C ^y C^, P, , P^, and P, are quantities which can be ob- 

 tained from Eqs. (76) through (78) and Y6(x) is given by Eq. (79). 

 We can see from the above equation that the far-field outgoing waves 

 are made of four independent wave components. The ternns other 

 than the one associated with Yg represent beating phenomena with 



939 



