Nonsinusoidal Oscillations of a Cylinder in a Free Surface 



VII. NUMERICAL RESULTS 



A semi-circular cylinder of unit radius (b = 1) is chosen for 

 sample calculations of the pressure distributions, hydrodynamic 

 forces, and out-going waves. The inputs for the calculations are the 

 values of the fundamental frequencies a, and o--. Three values of 

 (T, are chosen such that the corresponding length of gravity waves 

 in deep water, \| = 2irg/cr,^, are equal to 2b, 10b, and 20b. For 

 each value of o"| the values of the second frequency (72 are chosen 

 so that the wave lengths obtained by X2 = 2Trg/cr| lie in the interval 

 of \^<\2< 2X,| which is equivalent to 2(bo-|Vg) < b(r2/g < bo-.^/g. The 

 reason such a narrow range of o-g is chosen stems from our practical 

 interests. When the forcing frequencies are close the difference 

 between any two frequencies is very small. Thus any hydrodynamic 

 force response associated with the difference-frequency within this 

 band can often be treated as a d.c. force in practical situations. It 

 is of interest to examine how significant the magnitude of the hydro- 

 dynamic force of difference-frequency is compared with the pure d.c. 

 force. 



Numerical results are obtained here for the d.c. component 

 of hydrodynamic pressure, p , hydrodynamic force , f , and for 

 those quantities which are associated with the difference frequency, 

 p , f , and Y_ which represent outgoing waves at |x| = 00. The 

 quantities associated with the sum-frequency, Pg, f , and y , are 

 not computed because these quantities can often be approximated 

 from the known values associated with the frequencies Z<y^ or 20-2 

 when (T, ~ (T^- For comparison purposes the quantities Pj, f j, and 

 yj for i = 2 and 4 which are borrowed from Lee [ 1968] are also 

 shown with the quantities presently calculated. 



The deep-water gravity wave length based on the difference 

 frequency is given by 



K,= -^S.-,= K, 1^ 



where 



((T, - <y^)^ 1 +(X A) - Z^I}r/\ 



X. =.2Tr| ^^^ ^^^2^ 



0-. 0- 



Xg/X, as a function of X.2/^1 ^^ shown in Fig. i. This figure shows 

 the wave length corresponding to the difference-frequency (y^ - (Tp 

 compared to the fundamental wave lengths X.| and X.2* 



In the rest of the figures the abscissas are X which is defined 

 as \ = Xg/Xi- For the values of X, = 2b, 10b, and 20b, the corre- 



929 



