TRANSIENT OSCILLATIONS IN ELECTRIC WAVE-FILTERS 33 



and the appropriate mathematical procedure is essentially the same 

 for all the band pass wave-filters. 



The first method of solution outlined above for the low pass and 

 high pass filters, leads, for the L\C\LiCi type of band pass filter, to 

 the definite integral formula 



w 2 /•""/^ sin gx 

 A n {t)= — r— / : — cos2«/x-cos(y sin/i)<//i, (3.1) 



C0 m & 7T Jo g 



where x = cc m t; y = wt,'2; p = w/2u m ; and 



g = Vl+P 2 sin 2 /x- 



In solving this definite integral, use is made of the known formulas, 



•7T/2 



2 r lr / 2 



J2n(v) = — I cos 2wp-cos(v ship) Jp (3.2) 



7T J() 



and 



d- s 2 r w/2 



( — 1Y ^-z- Jtniy) =— I sin 25 p- cos 2wp -cos("y sin p)dp. (3.3) 

 ay" t Jo 



If in (3.1) g is replaced by unity, it follows from (3.2) that, to this 

 approximation 



w 

 A„(t) = — r J 2n (y) sin x (3.4) 



which is formula (3a) of the text 28 . Clearly this becomes an in- 

 creasingly good approximation as the parameter p becomes smaller; 

 that is, as the ratio of the band width w/2ir to the mid-frequency 

 co m /2ir becomes smaller. The approximate formulas of the text for 

 the other types of band pass filters were derived by precisely similar 

 procedure and involve approximations of the same character and order 

 of magnitude. 



To investigate the approximate solution, we proceed as follows: 

 If we write 



u?nk . , aS „ , v 2 r 1 */ 2 sin gx n , . . , ._ „ N 



A n (t) = F n (x,y) = — / — cos 2tifjL • cos(v sin n)djj., (3.5) 



w ir Jo g 



Gn\x,y) — —I sin gx • cos 2nn • cos(y sin n)dn (3.6) 



7Tt/n 



and 



and if we substitute for 1/g in (3.5) the expansion 



1 1-3 



1 - -p 2 sin 2 p + — p 4 sin V - • • • • 



2? If a series resistance Ri and a shunt resistance R 2 = k' i /R l are included in the 

 filter sections, the formula becomes (3.4) multiplied by the factor exp( — Riy/2k). 



