Nonsinusoidal Oscillations of a Cylinder in a Free Surface 



latlonshlp between the forced motion and the pulsating singularities 

 at the origin, and 



2 



' g 



Kj^2 ~ *" » ^o^ ^ ~ 1,2. 



We can also show from Eqs, (43) through (45) by letting |x| — ^ oo 

 (or \ — ^ CO for a = or - ir) that 



,00 



G,(x,0)-.Q,e-"' llm (C £ls-^ dp + J.e"'' cos K^x)| 



j(Kilxl-q:) 

 = - JTrQje ' ' for i = 5,6 (73) 



where 



=^ g 



K - (tri ~ 0-2) 

 6 g 



and then Eqs. (38) and (39) can be used to show that 



W,(x.O)~ j \ h(e)e * de, (74) 



J-oo 



i{(K,-K2)lxl-^} „a. ,,„.^, 



We(^.0)- ^"\,.K,.K,^ ii,M4)a ^ dS, (75) 



where a© and P are defined in Eqs. (30). The far-field behavior 

 of the derivatives of the functions <p\ (1 = 1,2,3,4), G; (1=5,6), 

 Wg, and Wg can be shown to differ from those exhibited in Eqs, (72) 

 through (75) by factors of the appropriate wave numbers Kj. If 

 these results are applied to Eqs. (67) through (71) and some mani- 

 pulations are carried out, we can show that as | x | — ^ oo 



Yi(x)~|Qiaie^^*'*'"""^ for 1=1.2, (76) 



927 



