1134 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



Bending effects are caused by locally large curvatures, and are significant 

 mainly where the cable leaves the pa3^-out sheaves and at the ocean 

 bottom. However, for a cable with a steel strength member, bending 

 even to the small radius of the pay-out sheave typically does not ma- 

 terially reduce the tension required to break the cable. Hence, in these 

 cases we can expect an analysis based on the first idealization to give a 

 reasonable idea of when cable rupture will occur. In laying, ship speeds 

 are normally steady and, with the exception of the fluctuations caused 

 by wave action which we consider later in the paper, the second idealiza- 

 tion is reasonable also. In recovery, on the other hand, ship speeds are 

 apt not to be steady, and the second idealization is more tenuous. But 

 because of the very slow speeds usually employed, this idealization may 

 in fact be meaningful in recovery as well. 



III. TWO-DIMENSIONAL STATIONARY MODEL 



3.1 General 



Assume that the cable ship is sailing at a constant horizontal velocity, 

 that the cable pay-out or haul-in rate is constant, and that the drag of 

 the water on the cable depends only on the relative velocity between 

 the water and the cable. Further, assume that in a frame of reference 

 translating with the ship the cable configuration is time-independent 

 or stationary. This idealized model of the cable laying or recovery pro- 

 cess we call the two-dimensional stationary model. 



This is the model which has been considered in the previous analytical 

 studies. ^'^^ As the early investigators quickly pointed out, when the 

 tension at the bottom of the cable is zero, the cable, according to this 

 model, can lie in a straight line from ship to ocean bottom. During lay- 

 ing, when slack is normally paid out, the zero tension condition actually 

 occurs, and hence this case is of considerable practical importance. 



The straight line can in fact be shown to be the only solution which 

 can satisfy all the observed boundary conditions. This point is discussed 

 in detail in Appendix A. That the straight line is a possible configuration 

 can be seen from Fig. 1. In the vector diagram the velocity of the water 

 with respect to the cable is resolved into a component Vn normal to the 

 cable and a component Vt tangential to it. Associated with Vn and Yt 

 are normal and tangential water resistance or drag forces Dn and Dt ■ 

 In the straight line configuration, the cable inclination is such that Dn 

 just balances the normal component of the cable weight forces. The 

 situation is thus analogous to that of a chain sliding on an inclined plane, 

 with the forces Dn corresponding to the normal reaction forces of the 

 plane. Summing forces in the normal direction, we get, thei'efore. 



