APPENDIX B 



THE DRAG ON THE ARRAY 



The problem formulated in Section IV is linear except for the term of 

 the drag on the array. This term makes the entire problem nonlinear, and 

 enormously more difficult than a linear problem. It is, therefore, expedient to 

 linearize this term. 



For sinusoidal inputs, one can take the drag on the array, D , as 

 being proportional to the velocity of the array (not the square of the velocity) 

 and then determine the constant of proportionality experimentally. For veloci- 

 ties and arrays of the type involved in this problem, this constant of proportion- 

 ality will be found to be a function not only of the frequency but also of the am- 

 plitude of the velocity of the array. However, such an experiment is not prac- 

 tical in the present case; no effective scaling, according to the principle of 

 similarity, of the variables is possible, because of the enormous dimensions of 

 the arrays and the appreciable velocities involved. We therefore propose the 

 following analytical linearization. 



The drag D on the array is given by (see Equation 8): 



D = - a p A 

 a I 



t (B-I) 



For sinusoidal inputs, it is expected that the motion of the array will be periodic. 

 If the I Su/Bt| is replaced by a constant, then D^ and the entire problem become 

 linear, and therefore the displacement of the array will be sinusoidal of ampli- 



Q 



tude U . We set the constant which replaced |Bu/St| equal to -r — ti)U. . This 



selection results in the same amount of dissipation of energy by the array when 

 u is taken as sinusoidal in both factors in Equation B-1 and when only the sec- 

 ond factor is taken as sinusoidal. Thus, with Uj^ a real positive number, the 

 amplitude of the drag D will be given by: 



D = ^apuj^AU,^ (B-2) 



a on 1 



34 



artliur 2I.K.ittk3lnr. 



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