1142 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



Similarly, the longitudinal sinking velocity v^ was the terminal velocity 

 of a cable length sinking with its axis constrained to be vertical. If for 

 a given cable the drag forces are functions only of velocity, the pa- 

 rameters w, lis , and Vs , together with the laws of variation of the drag 

 forces Anth velocity, completelj^ define the hydrodynamic behavior of 

 the cable. Since sinking velocities are still used in submarine cable tech- 

 nology, it is of interest to relate them to the more modern drag coeffici- 

 ent viewpoint. 



In the case of transverse or normal flow around the cable, the variation 

 of Z),v ^^'ith the square of the relative transverse velocity gives (Fat/Ws)^ = 

 Dx/w, since at a transverse velocity equal to the sinking velocity the 

 unit transverse drag force is w. Substituting for Djv from (4) we find 



Thus, the transverse sinking velocity Us is identical with the hydrody- 

 namic consant H. We can therefore alternativelj^ write the approximate i 

 relationship (9) as 



aoV = Us, (17) 



where ao is in radians and iis and V are in knots. 



For the tangential or skin friction flow along smooth cable, the sinking 

 velocity concept is inadequate because the unit tangential drag force < 

 Dt varies \nth length as well as ^^^th the relative tangential velocity *■ 

 Vt . However, for cable with the conventional jute exterior (cable Xo. 

 2), we have (Vf/Vs) ' = Dt, w and from (15) the vertical sinking velocity : 

 Vs is Vs = (46. Iw)^'^'^^, where Vs is in knots. j 



We note in passing that the cable does not, as is sometimes supposed, I 

 sink vertically to the bottom at the transverse sinking velocity Us ■ '< 

 Actually, the term "vertical cable sinking rate" is ambiguous. There are i 

 in fact two vertical sinking rates which maj^ be important. Although 

 both these rates are normallj^ approximately equal to Us neither is identi- 

 cal to it. 



Relative to the earth, the resultant velocitj^ Y n of a cable element ha- 

 two components: a horizontal component of the magnitude of the ship 

 \'elocity, and a component inclined at the angle a of the magnitude of 

 the cable pay-out rate Vc . These are shown in Fig. 7. The component 

 Fvert of V K , given by T^v<rt = F,. sin a, is the rate at which a cable 

 element sinks vertically. For a laying depth h, the time r it takes for ;i 

 cable element to sink to bottom is therefore r = h/Vc sin a. This time 

 would, for example, tell one how long it takes a lightweight repeater, 

 integral with the cable, to reach the ocean bottom. 



ij 



