1146 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



and hence the tension at the ship Ts is very nearly 



T. = To + wh, (21) 



where h is the depth at the touchdoAMi point. Thus, if the tangential drag 

 force is neghgible, the tension at the ship is essentiaUy the bottom ten- 

 sion plus wh, regardless of the nature of the normal drag forces. This is 

 in fact a form of a well-known theorem which, as we shall see in Section 

 7.1, apphes in the three-dimensional case as well. 



In the next sections we make further simplifications of the general 

 solution for the specific cases of laying and recovery. 



3.6 Approximate Solution for Cable Laying 



On long cable lays ship speeds are normally of the order of 4-8 knots, 

 with accompanying values of the critical angle a of the order of 10°-30°. 

 For these small values of a, the assumption of zero tangential drag to- 

 gether with some mathematical approximations allow further simplifi- 

 cations of the general solution. These simplifications are derived in de- 

 tail in Appendix C; here we indicate the results. The angle 6 which the 

 configuration makes with horizontal is closely given by 



1 - [%/{% + y)Y '^"' 



tan - = tan - 



l-f['ro/(n + ^)rtan^| 



(22) 



where y and To are dimensionless depth and bottom tension defined by 



y = y/h, 



To = To/wh. 



Here we use the cable angle a in the sense of Section 3.2, namely, as a 

 parameter characterizing the hydrodynamic cable properties and the 

 ship speed. The constant y is in turn defined by 



= (2 - «i"' ") . (23) 



sm^ a 



For small a, tan (a/2) is negligible and y is large. Further 



° < ?rh < ^- 



Hence, the denominator in (22) is very nearly unity and 6 approaches 

 the critical angle a at small values of y, even for relatively large values 

 of To of the order of three or four. Thus in the laying case, the cable 

 configuration is very close to a straight line except for a short distance 

 at the ocean bottom. 



