1188 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



zero. We write 



Po + 1\ = am, 



Qo = hg{t), 



where J{t), g{t) are some bounded functions of time and a and h are 

 constants. We assume that for given f(t) and g{t), T, p, and q vary 

 analytically with e, namely 



T = To + eTi + e'T2 + • • • , 



q = f gi + e'^q-i + • • • , (83) 



p = Po -\- epi + ep2 + • • • , 



with counterparts for the submerged cable. The stationary transverse 

 deflection is further assumed zero, and therefore the series for q contains 

 no go term. Substituting, for example, (83) into (82) for air and equating 

 like powers of e, we find 



(84) 



Equation (84a) of this sequence shows that only longitudinal displace- 

 ments are associated with stationary tensions, while (84b) indicates that 

 for small ship motions cable tensions are independent of the transverse 

 component of ship motion. To compute the effect of transverse motion, 

 (84c) shows that terms of the order e' in p and e in q must be considered. 

 We assume further that 1 + po^ ^ 1, since pox is the order of magnitude 

 of a strain. 



Equations (83) and (84) substituted into (76) yield with this approxi- 

 mation 



Wa COS a = 0, (a) 

 _ (85) 



EA Poxx - pa Pott + Wa slu CC = 0, (b) 



Qixx — —^qut =0, (a) 



Pixx -, PiH = 0, (b) (86) 



P'lxx ^P2tt = —^ qitiqixx — gix^ixx , (c) 



