IV. THE EFFECT OF MOTIONS OF THE SEA SURFACE 



A. FORMULATION OF THE PROBLEM 



A realistic wavy sea will cause the vessel to move in the vertical 

 direction, and, thus, generate waves in the cable which will interact with the 

 array This Section deals with the analysis of the dynamic stresses induced 

 in the cable by such waves . 



This problem can become as complicated as one wishes to make it. 

 Here we will aim at a reasonable linear analysis. Therefore, we must, first 

 of alL assume an elastic cable. Metallic cables remain elastic as long as the 

 total stress does not exceed their elastic limit. For the cable considered in 

 Section II, the elastic limit is about 1.5 x 10^ psi. The maximum static 

 stress calculated for the cable is below this elastic limit, and therefore, this 

 cable is still elastic. Nylon type ropes are more plastic than elastic, unless 

 they have been prestressed sufficiently. On the other hand, the dynamic stress 

 wave propagating in the cable can be a compressive one. Then, if this com- 

 pression exceeds the static tension in the cable, the cable will buckle, and the 

 propagation of stress in the cable becomes a nonlinear problem. Buckling is 

 treated briefly in Section VI. Since the present linear analysis assumes that 

 the cable does not buckle, any results which indicate that the cable should have 

 buckled are not valid. 



Consider the cable and the array in a vertical position. It has been 

 shown in Section III that the deflection of the cable by a reasonable horizontal 

 current is small under conditions stated there; therefore, even in the presence 

 of such a current, the cable can be considered to be vertical for our present 

 purpose. As the upper end of the cable moves vertically, waves will be propa- 

 gating in the cable. If u(x,t) is the dynamic displacement at time t of an ele- 

 ment of the cable A x , which in the absence of waves is located at x (see Fig- 

 ure 1), the governing equation for the propagation of waves is: 



5^u _ a 2 u Bu 



"c " 3t» ~ ^ ^ b^ " ^ dt 



p„ S ^ = S E ^ - K ^^ (4) 



10 



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