Nonsinusoidal Osoillations of a Cylinder in a Free Surface 



which is the same expansion as that assumed by Lee [ 1966] . We can 

 easily establish the identities <p^ - <P2 and ^3= <p^. It will now be 

 shown that for a = cr we also have <p = 2^ and q> = <p . Equation 

 (20) gives 



- 2(fl'u?'2x + ^iy^2y){ " J l| | ^2^^iyy " ^i^i^iy^ 



- 2(<P,x?'2x + ^'ly-Pzy^ I ' 

 so for o-j = 0" 



Comparison of this with Eq. (A- 2) shows that 



hgix) = 2h2(x) 



Equation (22) gives 



. b 



"^S^^o'^o) = - J 2 l^VK'^o^ ^ V^^'K^ - ^lyy " ^2yy [ 



so for (T. = (T 



m, = - jb 5 <p. (x ,v )f '(x ) - i . 

 5 '' ( ^ixy^ o''o' ^ 0' ^lyy f 



Comparison of this with Eq. (A- 5) shows that mg = 2mL,. The far 

 field conditions and the symmetric -flow condition for both <p- and 

 ^_ are essentially identical. The above results lead to the conclusion 

 that <Pq = 2^?,. <pe can be shown to be equal to <p by a similar proof 

 if h-(x) (Eq. (25) for a■^ ^ rf-g) is compared with h^x) (Eq. (A-3)) 

 and mg(x) (Eq. (27) for (r ^ = a.) is compared with m (x) (Eq. (A-6)). 



943 



