DYNAMICS AND KINEMATICS OF SUBMARINE CABLE 



1175 



APPENDIX A 



Discussio7i of the Tivo-Dimensional Stationary Configuration for Zero 

 Bottom Tension 



We assume again that the tangential drag Dt depends only on the 

 relative tangential velocity Vt , and we consider in a. T, 6 plane the solu- 

 tion trajectories of equations (18). These trajectories satisfy the ec^uation 



dT _ (sin d - Dt/w) 

 'dd ~ 



{T - pcV.'), 



(53) 



cos ^ — A sin I sni 6 



and are periodic in 6 ^^dth a period of 2ir. In Fig. 30 we have plotted the 

 solution trajectories qualitatively for {a — ir) ^ ^ (a + tt). It is seen 

 that the trajectories are either the vertical straight Hues d = a, 6 = a±w 

 or they lie completely within one of four regions, labelled I, II, III, or 

 IV, which are bounded by these vertical hues and the horizontal hne 

 T = pcVc. The trajectory 6 = a corresponds to the straight-line laying 

 configuration, while the trajectories 6 = a ± tt correspond to Shea's 

 straight-line recovery method. 



Examine now the trajectories in Regions II and III at a point of 

 which r = 0. As J. F. Shea has pointed out, these trajectories all he 

 below the line T = pcVf. On the other hand, the trajectory 6 = a con- 



(X+7T 



Fig. 30 — Qualitative representation of the solution trajectories of the two- 

 dimensional stationary model. 



