TRANSIENT OSCILLATIONS IN ELECTRIC WAVE-FILTERS 41 



This approximate formula is valid only where x>n, its accuracy 

 increasing with x and with n. For all orders of n it is quite accurate 

 beyond the first zero of the function. 



The "instantaneous frequency" of oscillation is approximately 



1 n'f \ 1 li m2 



3 m 2 



2x 4 (l-m 2 /x 2 ) 2 ' 



By this it is meant that at any point x (>«) the interval between 

 successive zeros is approximately ir/Q,'{x). Otherwise stated, in the 

 neighborhood of any point x, the function behaves like a sinusoid of 

 amplitude B„(x) and frequency u/2ir where co = Q w (a:). 



The following approximate formulas, while not sufficiently precise 

 for the purposes of accurate computation except for quite large values 

 of x, clearly exhibit the character of the functions for values of the 

 argument x>n, and of the order n>2. 



J n (x) = h nx j — cos (q„x-Q n ), 



\ 1TX 



J' H (x) = -q„h„ x l— sin (q„x-e n ), 



\ TTX 



r 



•> 



: }i n f~2 



J n (x)dx = l-\ x /_sin (q n x-Q„), 



2 \ TTX 



where 



and 



" \l-n*/x 2 ) + 4x 2 ' 



q„ = Vl-n 2 /x 2 , 



_ 2n+l . 



6» = —7 — ir — n sin l (n/x). 



Appendix III 

 Building-Up of Alternating Currents in Wave-Filters 

 The integrals 



S n (z\v)= / J„(zi) sin v(z-z l )dz 1 



and 



C n (z;v) = / J n (zi) cos v(z-zi)dz u 

 Jo 



on which the genesis and growth of alternating currents in the low 

 pass and band pass filters depends, have been computed as follows. 



