DYNAMICS AND KINENATICS OF SUBMARINE CABLE 1181 



Let the origin of an x, y coordinate system be at the cable touchdown 

 point (Fig. 8). Further, let x be the x coordinate of a point along the 

 cable configuration and s the corresponding distance along the cable 

 from the origin. If we define 



^ = s - X, (62) 



then 



dA ds dx ^ 6 ._„. 



and 



^ = sin 9. (64) 



as 



By means of (63) and (64), (61) transforms to 



.rp , .. dA d'A 1 {dAy (dA)' 1 „ ..,. 



dy dy'- 4 {dy) (dy) 4 



where we have in addition introduced the non-dimensional variables 



To = To/wh, 

 A = A/h, 



y = y/h. 



Here h is the ocean depth at the touchdown point. Using the condition 

 that 6 = a,t y = 0, which implies dA/dy = at y = 0, we get upon 

 integrating (65) 



dA ^ j^_^ a I 1 - [%/(% + yW Y" 



where 



2 - sm a , . 



7 = ^, . (67) 



sin^ a 



The usual range of the critical angle a is between 10 and 30 degrees. Also 



To 

 < = ■ < 1 



- T, ^y= ' 



Therefore, we approximate the denominator of equation (67) by unity. 



