Write the equation of motion (A-1) in the form; 



TM NOc 377 



k^^ ^J^ -- ''('■'^ ' (-1) 



where u(t) is considered to be a quas i -random function of time.' The functions .on <iD(t) 

 and u(t) may be written as the complex Fourier integrals; 



and 





'^ l-Jt 



n*>-[y «=/Z<'*^- (A-as) 



The term V represents angular frequency, and dW (V) and 42 (-t/) are increments 

 in the quasi"=random complex functions v^ (•u^) and ■g (V)» These functions have 

 the following properties : 



if- 



^ir, z. di/i = J\/ -* 



o 





Also: 



and 



jU^ dw ('\^)d(/^h) -nj/j. (A-25) 



c/v -*Q — """ — ~ " I Cv ) 



The asterisk indicates the complex conjxigate^ and "f^^) and $^-vj represent the 

 energy density of the spectrum functions of 6i>(t) and u(t), respectively (see 

 Batchelor, 1953). 



I Substituting equations (A-22) and (A=23) into (A=21), and remembering that 

 c/v^ (i;) aiidc/^(I»j are independent of t, the result is: 



A-.5 



