B. THE STEADY STATE SOLUTION FOR SINUSOIDAL INPUTS 



We will take u at x' = as sinusoidal; of amplitude U , and of 

 angular frequency u) and try to find the steady state solution of the problem. 



In Appendix A it is shown that, under conditions which are valid 

 here, the friction on the cable is proportional to the velocity, and an expres- 

 sion for the friction constant ( 0^, ) is derived. In Appendix B it is shown how 

 the drag term in Equation 10 can be linearized and how good such an approxi- 

 mation is. Once this linearization is introduced, the entire problem is linear 

 and therefore, for sinusoidal inputs we will have sinusoidal waves . 



It must be emphasized, once and for all, that the approximation of 

 the nonlinear quadratic drag on the array, developed in Appendix B, is not 

 linear but "quasi-quadratic." It replaces the drag, which is actually propor- 

 tional to the square of the instantaneous velocity, not by a term proportional 

 to the instantaneous velocity but by a term which varies in time as the velocity 

 and whose amplitude is proportional to the square of the amplitude of the vel- 

 ocity. 



In what follows a capital letter will be used to denote the complex 

 amplitude (magnitude and phase) of the respective instantaneous sinusoidal 

 variable. For instance, U(x') means the complex amplitude (magnitude and 

 phase) as a function of x' of the instantaneous sinusoidal dynamic displace- 

 ment u(x' , t'). 



Let us define a normalized displacement amplitude U' as being 

 equal to U divided by IUqI . If U^ is the value of U' at the array (x' = 1), 

 then the solution for U as a function of x' is: 



U = U cos («'y' + C sin ua'y' (11) 



where C is a complex constant and 



X , (JUL -,„. 



y' = 1 - x' = 1 - - , (w* = — (12) 



This solution satisfies Equation 7 if the friction of the water on the cable is 

 neglected. As is shown in Appendix A, this is valid for the frequencies of in- 

 terest here. Only at very sharp resonant peaks can the friction on the cable 

 be important, and even then its role can be insignificant if damping of a greater 

 order of magnitude exists somewhere else in the system (as, for example, in 

 the array). 



13 



S-7001-0307 



