DYNAMICS AND KINEMATICS OF SUBMARINE CABLE 1177 



In general, this upper bound will be much less than the laying depth. 

 For example, for cable No. 2 being laid with 6 per cent slack at 6 knots 

 ?/,„ < 12.5 feet. That is, the cable configurations corresponding to Region 

 II do not reach the ocean surface. Hence these solutions of the stationary 

 model do not in general satisfy all the required boundary conditions and 

 can be discarded. 



Similarly, in Region III, the laying trajectories for which To = are 

 in the range a < da ^ tt. Consider those for which ^o < 7r/2. We get for 

 these trajectories 



Pcv: n j sin ^ 



• "'" w Jeo \(A sin^ ^ — cos ^) 



X exp 



f^ w sin 7] — Dri-n) 7 

 'flo w (A sin^ 7/ — cos 77) 



where 1/^/2 is the value of ^ at = 7r/2. Let m be the minimum value of 

 sin 7] — {Driv)/'^) in the range a < rj ^ t/2. If, as in the usual case, 

 m is positive, we can obtain an upper bound on ij^/2 by replacing 

 sin 1] — {DT{'n)/w) by m in the right-hand side of (55). By this means 

 we find that 



^ PcVc 2(1 + cos' a) 

 yw/2 < 7 as — • 



w m tan a/2 



For cable No. 2 being laid with G per cent slack at 6 knots this relation 

 yields ?/x/2 < 1,100 feet. So in the usual laying depths, which are many 

 times greater than y^rii , the configurations in Region III for which 

 To = correspond to a value of 6 at the surface greater than 7r/2, or to 

 cable being paid out in front of the ship during laying. It is doubtful 

 whether such configurations would be stable and, at any rate, doubtful 

 whether cable would ever be laid in such a manner. Hence, we conclude 

 that these To = solutions of Regions II and III will in general be 

 mathematical curiosities, and that the only realistic laying solution of 

 the stationary model for which the bottom tension is zero is the straight 

 line 6 = a. 



APPENDIX B 



Computation of the Transverse Drag Coefficient and the Hydrodynamic 

 Constant of a Smooth Cable from Published Data 



From (6) of Section 3.2 we obtain the relationship 



