DYNAMICS AND KINEMATICS OF SUBMARINE CABLE 1205 



upper, cross-current stratum extended all the way to the ocean bottom. 

 That is, we take 6 and \l/ to be of the form 



6 = a' ^ 6, 



(108) 

 ,/. = ,/,, 



where a' is the stationary incidence angle corresponding to the velocity 

 V, and 6 and ^ are assumed small compared to unity. 



Substituting (48b), (107), and (108) into (47a, b) and retaining only 

 linear terms in 6, \p and their derivatives, we get the linear first order 

 equations 



riff 

 {h + 7?) -f + (2ctn'a' + 1)^ =0, (a) 



drj 



(109) 



(/i + ^) # + csc'a';/. = 0. (b) 



dr} 



Because in the lower stratum the cable is a straight line parallel to the 

 path of the ship, we have as boundary conditions: 



,= -/.',(^. = «-"'' (110) 



where h' is the depth of the upper, cross-current stratum and a is the 

 stationary incidence angle corresponding to the velocity V. 

 The solution of (109) for the boundary conditions (110) is 



^^ + '^ (111) 



- fh-h'X 



where 



M = (2 ctn'ct' -[- 1), 



V = cscV, (112) 



Aa = or — a\ 



Equation (48) for the space-coordinates ^, r;, and f of the cable in turn 

 can be written to terms of first order in the form 



-f = etna — 0CSC a , 

 drj 



(113) 



-— = — ,A,ctnQ: . 

 dj] 



