1172 THE BP]LL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



surface, and if at the bottom (77 = —h) the tension is zero, the tension 

 at the ship is essentially wh, regardless of the nature of the normal 

 drag forces. Since in most laying situations for present cables, the tan- 

 gential drag force can be reasonably neglected, this fact provides a con- 

 venient over-all check on the laying process. That is, if the cable is being 

 laid with excess, the tension at the ship for any stationary cable con- 

 figuration, planar or non-planar, should be essentially wh. Any marked 

 increase of tension over the wh value necessarily means the bottom ten- 

 sion is non-zero and insufficient cable is being paid out. 



The second important result is derived in Appendix F. This result is 

 that if the bottom ocean layer in our model is devoid of cross currents, 

 and if the bottom tension is zero, then, for the boundary conditions which 

 are normally observed, the cable configuration in the bottom layer is a 

 straight line. Further, this straight line is in the plane formed by the 

 ship's velocity vector F and the gravity vector. Hence, for example, in 

 laying with excess in a sea which contains surface currents, the cable 

 configuration in the lower, current-free portion will be a straight line in 

 a vertical plane parallel to the resultant velocity of the ship. The laid 

 cable will be parallel to the ship's path, but displaced a certain distance 

 from it. Thus, because the lower portion is a straight line, our previous 

 results about the kinematics of straight-line laying still apply. Only they 

 now are pertinent to the displaced bottom contour rather than to the 

 contour which hes directly beneath the ship. 



7.2 Perturbation Solution for a Uniform Cross-Curreyit 



Cross-currents are commonly confined to a region near the ocean sur- 

 face. It is of interest therefore to determine for such surface currents the 

 distance e (Fig. 26) which the laid cable will be displaced from the path 

 of the ship. In Appendix F we consider the problem for a cross-current of 

 uniform but comparatively small velocity. In addition, we determine the 

 distance d (Fig. 26) back of the ship at which the cable leaves the upper, 

 cross-current stratum and assumes the straight-line configuration it has 

 in the lower stratum. Let us assume for the sake of reference that the 

 resultant ship velocity V is due east, and that the cross-current F„ is 

 inclined at an angle jS to the north (Fig. 26). The resultant velocity V 

 of the water with respect to the cable in the surface stratum has the 

 magnitude therefore of 



V = \{V - F„, cos ^f + (F, sin /3)']', (50) 



and is incUned at the angle ip from due \vest, where 



