118G THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



Summing forces along the directions t (tangential) and n (normal) shown 

 in Fig. 33, dividing by A.t (air) or Af (water) and letting Arc — > and 

 Af —> 0, we obtain the following ecjuations of equilibrium* 



Air: 



Tipx — Wa cos (e — (p) = pciqtt cos <p — ptt sin ip), 



Tx + Wa sin {6 - if) = pciqit sin (p + ptt cos <p), 



Water : 



T^f + Z)A'Sf - w cos (d - if) = paiqtt cos ip — ptt sin <p), 



T^ -\- w sin {d — (p) = pciqtt sin <p + qtt cos (p). 



(76) 



(77) 



Here, pc denotes the mass per unit length of the cable in air. As is known 

 from hydrodynamic theory, in order to accelerate a body through a 

 fluid, one must change not only the momentum of the body but that of 

 some of the surrounding fluid as well. Thus the body has a virtual or ap- 

 parent mass in addition to its intrinsic mass. In the first (77), the equa- 

 tion of equihbrium in the normal direction in water, we accordingly use 

 Pu- , given by 



Pw — Pc -T -i^a p 



as the intrinsic plus virtual mass per unit length of cable moving through 

 water. The quantities d and p are the outer diameter of the cable and 

 mass density of the water, respectively, and the quantity (T/4)rf'p is 

 the virtual mass of a unit length circular cylinder moving transversely 

 through water. 



We take for the normal drag force per unit length 



D^ =^ V^ I V^ |. (78) 



Here Vn is the normal component of velocity of the water relative to 

 the cable, i.e., 



Vn = V sin (6 — (p) -\- Vt sin <p — rjt cos <p, (79) 



and CDpd/2 is a constant. 



The quantities s and <p are given by the following geometric relations 

 which can be obtained from Fig. 33: 



* We use the subscript notation for differentiation throughout this section, 

 e.g., fx ^d'p/dx. 



