DYNAMICS AND KINEMATICS OF SUBMARINE CABLE 1191 



gi = Re Qi{x) e'"', 



Vy = Re i^i(f) e'"', 



where the factor e'"' will be henceforth suppressed. 



The solution of (86a) and (87a) for the steady state may then be 

 written 



^ . s p ioiX . j^ ( ioix\ 

 Qi{x) = Bi exp — + Bi exp I 1, 



Co \ C2 / 



^i(f) = Fi exp (gif) + F^ exp (g.r), 



where the -B's and F's are complex constants and gi and 52 are the roots 

 of the quadratic 



2 

 q — 8q — icoy + -^ = 0. 



C2^ 



Throwing aw^ay the root of this equation which corresponds to the in- 

 coming wave in water, we get 



/fi(f) = Fexp(g,f). 



where gi is the root corresponding to the outgoing wave. The three com- 

 plex constants Bi , B2 , and F can now be determined from (89a), (89b) 

 and (90) 



Bi -\- Bo = A/e, 

 B, exp — + 52 exp (^- _-j - /^ = 0, ^^^^ 



— Bi exp i52 exp ( — 



C2 L C2 \ 



C2 /J 



- qiF = 0. 



We note that Bi , B2 and F are proportional to the amplitude A of the 

 forcing motion. 



D.5 Second-Order Longitudinal Response 



From the preceding results, the right-hand sides of the equations of 

 longitudinal motion (86c) and (87c) can be computed. This computation 

 for (86c) results in 



