1170 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBEll 1957 



configuration is everywhere the velocity of the ship, which we denote by 

 the vector V- Hence the velocity V' of the water with respect to the 

 cable configuration in the reference layer is 



r = v^ + i-v) 



Vu 



V. 



Further, in this layer we choose a set of coordinate axes ?, r?, T translating 

 at the velocity V as follows: The ^ axis has the direction of — V', while 

 77 is measured vertically upward, and f is perpendicular to rj and ^ so 

 that the axes ^, ??, f form a right-handed system. A plan view of this 

 configuration is shown in Fig. 26. We have denoted the angle between V 

 and Vw by jS, while the angle between the ^ axis and V is denoted by (p. 

 (The distances d and e refer to a subsequent section.) To describe the 

 cable configuration with respect to the ^, r?, f axes, we use the spherical 

 polar coordinates 6 and i/' shown in Fig. 27. (The t ,u ,v vectors are dis- 

 cussed in Appendix F.) 



Fig. 26 — Plan view of the coordinate system for the three-dimensional sta- 

 tionary model. 



As in the two-dimensional case, we resolve the velocity of the water 

 with respect to a cable element in the reference layer into a component 

 Vn normal to the cable and a component Vt tangential to the cable, and 

 associate with Vn and Vt the drag forces Djv and Dt . The resulting 

 differential equations, which are derived in detail in Appendix F, are the 

 following: 



(T - p^^:) f 



as 



+ i:;A'(cos^ x// sin" 6 + sin" i/')"" cos i/- sin d — w cos = 0, (a) 



