DYNAMICS AND KINEMATICS OF SUBMARINE CABLE 1145 



At the cable touehdowTi point on the ocean bottom only two conditions 

 are possible. If the angled is not zero or tt there, the cable tension T must 

 be zero. Otherwise a finite tension would act on an infinitesimal length 

 of cable, producing an infinite acceleration. Hence, either the tension T 

 must be zero or the angle 6 must be zero or tt. The first case normally 

 implies a straight-line configuration (see Appendix A), which has al- 

 ready been discussed. In other cases, we define To as the tension at the 

 touchdown point, and we let ^o be zero or x, whichever is appropriate. 



If X, y are coordinates in the translating {x, y) frame of a point along 

 the cable configuration, then 



dx = ds cos d, 

 dy = ds sin 6. 

 Combining these relations with (18a), we have 



I 



y 



9q w(cos ^ — a sin f I sin ^ |) 



_ /•" {T - PeVc) COS ^ 



Jbq iy(cos ^ — A sin | | sin ^ |) 



_ [' {T - poV') sin ^ 



Jeo w{cos ^ — A sin ^ | sin ^ |) 



Equations (19) and (20) are an integral representation of the complete 

 solution of the basic two-dimensional model. In general, the integrals 

 appearing in these equations cannot be evaluated in terms of elementary 

 functions, and the solution must be obtained by numerical integration. 

 For towing problems where the pay-out velocity is zero, Pode " has tabu- 

 lated these numerical integrations using the approximation that Dt has 

 certain constant ^'alues. However, in towing problems the direction of 

 Dt is opposite to what it is in normal laying and recovery problems. Be- 

 cause small magnitudes of Dr were used, these tables nevertheless usually 

 give adequate results in laying and recovery situations as well. At the 

 same time, for submarine cable problems, other approximations allow 

 more convenient ways of evaluating the integrals of (19) and (20). 



For example, it is more accurate simply to assume that Dt is zero. As 

 we indicated in Section 3.2, this approximation gives a negligible devia- 

 tion from the exact solution if the relative tangential velocity Vt is small. 

 Furthermore, in this situation we obtain from (18b) 



dT . , dy 



-— = w sin 6 =^ w -f- , 

 ds ds 



