1162 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



inclined than the cable (a > 7), the rate of decrease in shipboard ten- 

 sion is 



dT sin a sin 7 ,7- 



at sm {a — 7) 



Like negative slack laying on a flat bottom, the variation of tension 

 with time depends greatly on the frictional characteristics of the bottom 

 in cases other than the above. We therefore limit ourselves to situations , 

 where the cable does not move with respect to the ocean floor. This case ! 

 might be approximated by rough bottoms, where the cable might wedge 

 itself between rocks. 



A nomograph giving a rough estimate of the rise of mean tension with 

 time when a cable becomes completely suspended is worked out in Ap- 

 pendix E. In deri^dng this nomograph it is assumed that the cable takes i 

 on a sequence of stationary configurations. This assumption is probably ] 

 reasonable if the time span of the tension rise is large compared to the - 

 time of passage of a tension wave from the ship to ocean floor and return, 

 which as mentioned in Section 4.1 is of the order of 18 seconds. However, j 

 because of this assumption and others mentioned in Appendix E, we | 

 regard the tension variation computed by the nomograph only as a crude 

 approximation. i 



Fig. 21 shows the mean ship-board tension versus time computed by 

 means of the nomograph for various slacks e, where e is defined by (31). 

 The values which were used for the other parameters entering the calcu- 

 lation were 



^^ = 3.1 X lO-l 



EA 



a = 12°. 



Also shown on this curve is the tension rise computed for the case of 

 laying down a vertical slope without excess. The rise for this case is given 

 by (38) with /3 = 90°. It is seen that as the slack e is increased the curves 

 for a complete suspension approach the |S = 90° curve. Indeed, it can be 

 shown that under the assumptions made in computing Fig. 21 the (8 = 90** 

 curve gives a lower bound on the tension rise with time in the case of a 

 complete suspension. A tension rise rate greater than the j8 = 90° rate 

 is thus an indication of unsatisfactory covering of the bottom. 



In the case of too rapid a ship speed resulting in a < 7 (Fig. 19c) re- 

 straint of movement of the cable along a rough bottom would cause the 

 tension on the high side of the crest to be zero. There would thus be a 

 sudden drop in tension corresponding to the sudden decrease in depth 

 at the touchdown point after the cable was laid over the crest of the hill. 



