1202 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 

 APPENDIX F 



The Three-Dimensional Stationary Model 



F.i Derivation of the Differential Equations 



Let i, j, k be unit vectors along the ^, ?/, f axes (Fig. 26) and t a unit 

 vector along the tangent to the cable configuration in the direction of 

 positive s. As in the two-dimensional model, we take this to be oppositr 

 to the direction of travel of the cable elements along the configuration. 

 With respect to the cable configuration the resultant velocity vector of 

 the water is in the —i direction. We resolve this velocity into directions 

 normal and tangential to the cable in the plane formed by i and 7. The 

 unit vector in the normal direction we denote by n, namely, 



n = - — . , ... ... , . (101) , 



I -« + (f Oil II 



In analogy to the two-dimensional model we assume the normal and 

 tangential drag forces depend only on the corresponding water velocity i 

 components. Thus, we take 



D, = ^Ci-nV)\ (102) I 



. I' 

 Equilibrium of the forces acting on a cable element yields the equation 



t'^+I^ + IDt^ nDs - }w = Pea. (103) !| 



ds ds 



The vector a denotes the acceleration of an element of the cable as it 

 moves at the constant pay-out velocity Vc along the cable configuration. 

 It is easily shown that 



a = V:' ^ . (104) 



ds 



For convenience we introduce a second reference triad of orthogonal 

 unit vectors?, u, and v as follows. The v vector is taken in the (^, f ) plane 

 normal to t ; the u vector is chosen equal to the vector product v X t- 

 The angles \l/ and d shown in Fig. 27 describe the orientation of the (7, 

 u, v) triad. In terms of these angles, we read from Fig. 27 the following 

 table of direction cosines 



-+ -^ -k 



i j k 



t cos 6 cos y}/ sin d —cos 6 sin yp 



u — sin d cos \p cos 6 sin 6 sin \}/ 



V sni yp cos xp 



! 



