1184 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



(74) 



e = at 1/ = 0, 



d = —as at y = h, 



where as is the incidence angle of the cable at the ship (Fig. 8). With 

 these boundary conditions, the development leading to (66) yields (26) 

 for the relationship between shipboard tension Ts and the incidence 

 angle as . 



To evaluate *S' and X, the values of s and x at the ship, we use (19), 

 (20a) and (20b). Again we simplify by assuming Dt = PcVc — 0. Fur- 

 ther, since for recovery — tt ^ ^ 0, we may write | sin ^ | = — sin ^. 

 This gives 



1 '•^ 



s = tJ 



X = 7'o 



COS ^ + A sin^ ^ 

 cos ^ 



exp 



/ 



Jo 



•^0 



sm 77 



cos r; + A sm'' r] 



iU, 



sm 7} 



dr) 



(75) 



dl 



_cos ^ + A sin^ ^ Jo cos ?? + A sin^ 77 



The dimensionless bottom tension 7'o is computed from (26). The inte- 

 grals appearing in (75) have been evaluated numerically. The results 

 are shown in Figs. 14 and 15. 



APPENDIX D 



Analysis of the Effect of Ship Motion 



D.i Formulation of the Differential Equations 



To analyze the effect of ship motion on cable tensions, we use the model 

 shown in Fig. 32. We assume the cable is a perfectly flexible and elastic 

 string whose motion is planar. The distance L along the cable from the 

 ship to the point of entry into the water is taken as constant, and the 

 longitudinal damping as negligible. 



I 



00 



W WW w W w w P^ 



WATER 



Qc 



AIR 



Fig. 32 — Model vised for the analj^sis of ship motion tensions. 



