1190 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



pi{0, t) = , (a) 



e 



P2{0, t) = 0, (b) 



?i(0, /) = Qo/e. (c) 



(88) 



That is, we apportion all of the longitudinal boundary motion to pi , 

 and all of the transverse boundary motion to qi . Equations [(84b), (86b) 

 and (87b)] then give the complete tension due to the longitudinal com- 

 ponent of ship motion to first order. As mentioned in the text, this ten- 

 sion is easily obtained from standard references, and is also the greater 

 part of the ship motion tension. 



To determine the tensions due to transverse ship motion, we solve 

 (86c) and (87c) for boundary conditions (88b) and (88c). In addition, 

 we have the transition conditions 



qi(L,t) = 7ji(0, 0, (a) 



gi. (L, t) = vh{0, t), (b) (89) 



P2 (x = L,i) = M^ = 0, 0, (c) 



P2X (L, t) = 7>2r(0, t), (d) 



which follow if we assume that at the point of entry into the water the 

 cable is continuous and the tensions are finite and continuous. 



We consider only the problem of the tensions associated with a har- 

 monic steady-state transverse disturbance. Ecjuations (86a) and (87a) 

 show the transverse response to this disturbance to be independent of 

 the longitudinal motion to fu'st order. The first-order transverse motion 

 in turn can be thought of as a forcing action on the second order longi- 

 tudinal motion, as (86c) and (87c) indicate. This suggests the program 

 we follow to compute tensions. Namelj', we first determine the first- 

 order steadj^-state transverse response, then the second-order steady- 

 state longitudinal response which is excited by the first-order transverse 

 oscillation, and finally, b}' (84c) the resulting tension caused by trans- 

 verse motion. 



D.4 Transverse Response 



At the ship we assume a harmonic forcing function 



Qo{t) = A cos o:t, (90) 



and we introduce complex exponential representation 



