DYNAMICS AND KINEMATICS OF SUBM.\RINE CABLE 



1159 



slack with respect to the bottom, the zero touchdown tension condition 

 is fulfilled. Hence, under the proper slack pay-out, the cable geometry 

 and, as we shall see, the cable kinematics are particularly simple. 



Essentially, we must consider two deviations from the horizontal 

 bottom, namely, downhill or descent laying and uphill or ascent laying. 

 We consider these situations in turn, confining ourselves to bottoms of 

 constant slope since any bottom contour can be approximated by 

 straight-line segments. 



To cover a descending bottom, the cable pay-out rate must exceed the 

 ship speed, Fig. 19(a). To cover an ascending bottom, the angle of in- 

 cidence a of the cable, which as we have seen in Sections 3.1 and 3.2 de- 

 pends only on the ship speed, must exceed the ascent angle 7, Fig. 19(b). 

 Otherwise, the situation shown in Fig. 19(c) develops. Hence the critical 

 parameters are pay-out speed and ship speed. 



(b) ASCENT (a > 7) 



(C) ASCENT {yxx) 



Fig. 19 — Cable geometry during straight-line descent and ascent laying. 



During descent laying we see from Fig. 20 that in a time / an amount 

 of cable equal to a + 6 must be paid out. Hence the required paj'-out 

 rate Vc is (a -f- h)/t. But by straightforward trigonometry 



y _ ct + 5 _ sin a + sin /? y. 

 " t sin (a + &) ' 



(30) 



where jS is the angle of descent and a is the straight-line incidence angle. 



