Lee 



is associated with the difference-frequency changes very slowly in 

 time and often may be treated as a pseudo-steady force. In fact we 

 can show that in the limiting case of a, = ^2. that the difference- 

 frequency force reduces to the steady-state force or in our notation, 

 fg = fy when cTi = erg- It is shown in Appendix A that for ^\ - ^ z 



Equations (55) through (57) show immediately that p. = p-, p,= p^ = 

 p_/2, and Pg = Py Substitution of these relations into £q. (64) leads 

 to fg = f Y for (T, = (Tg. We observe in Figs. 5, 6, and 7 that as 

 _\-^ 1.0, i.e. ""g "*■ °'i » Ufil ~^ I^tI and in Fig. 8 that yg"* -ir/2 as 

 \ -* 1.0. Since fgig. .^ = ffg| sin yg from Eq. (66) and i-j is nega- 

 tive in this case, we see that Yelo- = <r ~ ~ """/^ in order to maintain 

 the relation f g = fy for (r| = ag. 



For sufficiently small values of (r, - CTg. we can show that the 

 expression of the forcing motion becomes 



7/ = sin (T, t + sin Cgt 

 ho 



« 2 sin o-gt cos ^°" ' "^"" 2 ^ t (80) 



and the corresponding expression for the hydrodynamic force can be 

 derived from Eqs. (63) and (66) as 



F K €2 I f 2I cos ""' ^ ""^^ t sin (o-gt + Yg) 



+ € j4|f4| cos (a, - ^^t sin [Zfy^^ + y^) 



+ |fg| sin YgCos (o-, - (r^t + f 7 f + O(e^). (81) 



This is a beat oscillation for small values of cr, - (Tg. The response 

 of hydrodynamic forces to this beating motion is made of two kinds 

 of beat oscillations: a slowly-varying sinusoidal oscillation, and a 

 steady component. For comparison purposes the relative magnitudes 

 of the different components of the hydrodynamic force given in Eq. 

 (81) are shown in Table 2 for \,= 10b and \2=llb i.e. \=i.i. 

 The values in Table 2 are obtained from Fig. 6. 



938 



