DYNAMICS AND KINEMATICS OF SUBMARINE CABLE 1187 



sr = [(1 + vd' + ^r'],* (80) 



tan <p = 7,f/(l + m), (81) 



with similar expressions obtaining for the cable in air. 



The tension T in turn is given by the Hooke's Law or stress-strain re- 

 lation 



T = EA{[a + p.)' + q.'f - 1}, (air) 



., ,. (82) 



T = EA{[(l-{- vi)- + n/Y - 1}. (water) 



As we indicated in Section 4.1, we shall assume that the extensile rigidi- 

 ties EA corresponding to complete restraint and no restraint to twisting 

 will give the limiting values of the ship motion tension. 



Equations (76) through (82) form a complete system in terms of the 

 independent variables x or ^ and t. Formulating boundary conditions in 

 terms of the coordinate x (or f ) is awkward. This coordinate is measured 

 along the unstretched cable so that a disturbance applied at the ship is 

 applied at different a:'s as the cable is paid out. At the same time, if the 

 velocity of the pay-out is small compared to the significant wave velocity 

 of the cable then we can plausibly neglect the paying out effect. As will 

 be shown subsequently, in the problem at hand there are two significant 

 wave velocities, roughly corresponding to transverse and longitudinal 

 motion. The first of these is of the order of 200 ft/sec, while the second 

 is of the order of 5,000-10,000 ft/sec. On the other hand, the pay-out 

 velocity is of the order of 10 ft/sec. Hence, we take the pay-out velocity 

 to be zero. This allows us to use (76) through (82) without further trans- 

 formations and to identify x and f as coordinates fixed in the translating 

 reference frame. 



D.2 Perturbation Equations 



We assume that the motion is a small perturbation about the undis- 

 turbed configuration of our model. To determine which terms of the 

 differential equations are important in this case, we adopt the following 

 procedure. Let 



M = max[(Po + Pif + Q't 



where Pq and Qo are displacements of the cable at the ship, and Pi is the 

 variation of the pay-out displacement from the mean. The quantity 

 e = M/L will normally be less than unity, and for no ship motion will be 



