TRANSIENT OSCILLATIONS IN ELECTRIC WAVE-FILTERS 7 



since the integrand is everywhere positive, and this without any 

 explicit reference to the transient phenomena themselves. Formula 

 IX is of particular importance, because, as will be shown in a sub- 

 sequent part of this paper, it represents, except in limiting cases, the 

 relative amount of energy absorbed from random interference. 



III. The Indicial Admittances of Wave-Filters 



We are now in possession of the necessary formulas and mathemati- 

 cal processes for investigating the behavior of wave-filters in the 

 transient state. We shall first write down the indicial admittances 

 of the representative types investigated, their derivation being dis- 

 cussed in Appendix I. The formulas given for the low pass and the 

 high pass are exact, while those of the band pass filters are approxima- 

 tions based on the assumption that the transmission band-width is 

 small compared with the "mid-frequency" of the transmission band. 

 They are therefore formally restricted in their application to "narrow 

 band " filters. The analysis of the exact formula, given in Appendix I, 

 shows, however, that the deductions drawn from the approximate 

 formulas of the text are quite generally applicable without errors 

 of any practical consequence to band pass wave-filters, even when 

 the transmission band is relatively wide. These questions are fully 

 discussed in the Appendix. 



In the formulas given below the filters are assumed to be infinitely 

 long and the voltage to be applied at "mid-series" position to the 

 initial or zero-th section. A n {t) is then equal to the current in the 

 nth section in response to a unit voltage (zero before, unity after 

 time / = 0.) 



1. Low Pass Wave-Filter, Type L1C2, Fig. 1. 



A n {t) = \ £j 2n (x)dx, (la) 



where x = u c t, 



a> c =2/ v LiC2 = 27r times the critical or cut-off frequency, 



K. L, 



o 'WB'0'^ , ^ffifflP 



Fig. 1 



