To estimate the error involved in the above approximation, we consider the 

 array alone and a sinusoidal force acting on it (in the actual problem this force 

 is the force exerted by the cable) . If we denote the velocity of the array by v, 

 then the dynamics of the array will be governed by the following differential 

 equation: 



dv ^ 11 



-— = F cos n)t - p V V 



dt ^' ' 



(B-3) 



where F and p are constants . Then, according to the above approximation: 



V = V cos (u)t - (^ ) 



(B~4) 



where: 



= 2 



3ttu) Y 

 I6py 



1 + 



'\i 



( l6pF 



3 TTU) 



9 



2l^ 



(B-5) 



and (^ can also be computed. (In obtaining the above result, |vl in Equation 



g 



B-3 was replaced by -r— V.) This solution can be considered as a first iteration 

 •^ 3tt ' 



toward the exact solution. Now we can compute a correction v' through a sec- 

 ond iteration, by solving the equation: 



dv' , 8pV , 

 dt 3tt 



/ 8pV 



(B-6) 



where v and V are as given by Equations B-4 and B-5. By expanding the 

 term in the right hand side of Equation B-6 into a Fourier series, we find that 

 v' is given by a Fourier series, the frequencies of the components being 3(JD , 

 5uj , 7u) , . . .(i.e. , v' has no fundamental component!). This result is due to 

 the above choice of the constant replacing|v| in the first iteration. The am- 

 plitude Bo of the leading component of v' of frequency 3to is given by: 



Bg = 0.2 V 



1 + 12.5 



P V 



(B-7) 



This shows that the correction v' is not more than 20% of v, and, therefore, 

 the error involved in the above linearization of the drag is at most of the same 

 order. 



35 



%\\\\m a.littlc Jnt. 



S-7001-0307 



