1176 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



tains all values of T. Hence, according to the stationary model, the only 

 cable configuration for laying which has the value T = and values of 

 T ^ pcVc is the straight line inclined at the critical angle a. The mag- 

 nitude of pcVc is small. For example, for cable No. 2 paid out at six 

 knots PcVc is roughly six pounds, and for conditions approximating 

 stationary laying the observed tensions at the ship are in practice always 

 many times the pcVc value. For such magnitudes of shipboard tensions 

 and a zero bottom tension, the two-dimensional stationary model thus 

 yields the straight line as the only possible cable configuration. 



HoAvever, the empirical fact that T > pcVc does not guarantee that 

 the shipboard tension must be greater than pcVc . We might somehow 

 contrive to lay at a zero bottom tension with T < pcVc and ^Wth the 

 cable in one of the non-straight line configurations of Regions II or III. 



Consider the cable configuration lying in Region II. From Fig. 7 it can 

 be seen that the vertical velocity of a cable element is given by 



37 = - Fvert = - Vc sin d, 

 at 



where y is measured upward. Hence, of the possible trajectories for which 

 the bottom tension is zero only those for which the bottom cable angle 

 do is between zero and tt correspond to cable laying. For region II, there- 

 fore we need consider only the trajectories in the range ^ ^u < a at 

 To = 0. From (20c) the maximum value of y^ for these trajectories is 

 given by 



ft 



PcVc' n sin^ 



Vm = 



w Jo \(cos ^ — A sin- ^) 



'o IV sin 77 — Drir]) 



X exp 



w (cos 7; — A sin^ r?) 



dr) 



Let (-Dr)/^ be the maximum value oi Dt ,0 ^ 7? < a. With Dt set eciual 

 to (Z)7-)m , the right-hand side of (54) gives an upper bound on y^ • This 

 substitution further allows one to evaluate the right-hand side of this 

 equation in terms of standard integrals. The result yields the following 

 upper bound on y^: 



2 1 



y,n < 2.1 

 where 



P^ 



■Vc 



w 1 — r 



(Dr), 



w sm a 



