DYNAMICS AND KINEMATICS OF SUBMARINE CABLE 1161 



f = §. (34) 



Finally, using this expression for/ in (31), we arrive at* 



Ve - V = ^. (35) 



Thus, the increment in required pay-out rate is essentially a function 

 only of the descent angle 13 and is independent of ship speed. 



In the case of an ascending bottom for which a > y, Fig. 19(b), posi- 

 tive bottom slack may be obtained with a pay-out of less than the ship 

 speed. The allowable decrement in pay-out rate is given by 



V-Vo = ^, (36) 



that is, the same as the required increment for ascent laying. Likewise, 

 the fill / in this case is simply / = — {Hy/2V). 



The only way to avoid the situation shown in Fig. 19(c) where a < y 

 is to sail slowly enough to maintain an incidence angle a greater than 

 the angle of rise y. By (9), we have for most laying speeds aV ^ H. 

 With good accuracy the condition a > y thus implies 



V <-. (37) 



7 



Therefore, for a given rise y the limiting ship speed is simply H/y. 



5.2 Time-Wise Variation of the Mean Tension in Laying Over a Bottom 

 of Varying Depth 



In the cases where the cable is paid out with excess onto a bottom of 

 constant slope, the variation of the mean tension at the ship with time 

 is easily computed. During descent laying the increase in depth 8 after 

 a time t is by elementary trigonometry (Fig. 20) 



_ sin a sin /3 „ 

 sm (a -}- (8) 



Hence, the rate of rise of the mean shipboard tension is 



dT w8 sin a sin j8 ^r /oo\ 



— - = -- = ^— — ■ — r wV. (38) 



dt t sm {a + 13) 



Similarly, during an ascent lay for which the bottom is less steeply 



* Note that in (33), (34) and (35), // nuij- he replaced by the numerically identi- 

 cal transverse settling velocity Us (see Section 3.4). 



