DYNAMICS AND K1NE1VL\TICS OF SUBMARINE CABLE 



1193 



and 



/ = arg F, 



S f ^2w, 



g = arg (-gO, ^ g ^ -. 



It is seen that expression (92) and (93) have terms of the form 



sin 2(jot 



F{x)< 



cos 2(x3t 



(94) 



in addition to functions of x (or f) alone. In accordance with the idea 

 that the first order transverse motion is a forcing action on the second 

 order longitudinal motion, we take as solutions of (86c) and (87c) func- 

 tions of the form 



Gix) 



[sin 2cot 

 I cos 2ut 



to correspond to terms of the type given by (94) and functions of x (or f ) 

 alone to correspond to forcing terms which are independent of time. This 

 again gives linear differential ecjuations which can be readily solved. For 

 example, corresponding to the first term in (92) multiplying cos 2o}t we 

 have the assumed solution 



G{x) cos 2iot, 



and the differential equation 



(fG , 4a;' 1 



dx 



Ci 



This has the solution 



- 1 



3 r 



2coa; , 2w.r 



n sni + r-2 cos 



Co 



Co 



G 



cox 



, WJU , , . COX , CO 



Ai cos — + .42 sin — + 5— 



Cy Cl 0C2 



. 2cox , 



n sni h 1-2 cos 



C2 



2ux ' 



C2 /_ 



where Ai and .42 are undetermined constants. 



In this manner, the solution for the longitudinal motion can be ob- 

 tained in terms of a set of constants. These in turn can be evaluated by 

 means of the boundary and transition conditions on po . This evaluation, 

 although straightforward, is verj' tedious. We shall omit the details of 

 it here. The final result is 



