The two unknown constants U, and C will be determined by the 

 two boundary conditions. By defining the origin of the time (t') properly, we 

 can take U^ as being real and positive. Then, according to the linearization 

 introduced in Appendix B, substituting Equation 11 in Equation 10 we obtain: 



(UJ')^ U^ - ud'uC + i0(u)')2 (U')^ = 



(13) 



where: 



4a PA |Uol 



(14) 



3ttM„ 



This parameter is the ratio of the drag to the inertial force of the 

 array. Therefore: 



c = — u, ( - 1 + i euj 



u 1 r 



(15) 



Then Equation 11 reduces to: 



U' = u' sec (^ cos (UJ'y' + <:4) + i (U' )^ tan sin U)'y' (16) 



where: 



tan (4 = — , <<t) < - 



(17) 



Finally, the unknown real positive constant Uj^ will be determined by requiring 

 that |u'l at y' = 1 be equal to 1. This gives: 



(U')^ = cos («)'+0) 



2 B^ sin^ sin^ uu' 



1 + 



B^sin^ m' sin^ 2 

 cos* (tt)' +4) 



(18) 



If we denote the amplitude of the dynamic stress by Y. and define a 



normalized stress amplitude, E' , as being equal to 



Lr 



U E 

 ol 



, then the dis- 



tribution of E ' is given by: 



E* = («• u'^ sec sin (tt)'y' + <t>) ' ituB (u')^ tan cos UD'y' (19) 



14 



artliur JH.liittU.lint. 



S-7001-0307 



