TRANSIENT OSCILLATIONS IN ELECTRIC WAVE-FILTERS 29 



energy absorbed from random disturbances by a pure resistance net- 

 work is proportional to 



f R(a>)dt 



The relative amount of energy absorbed by the high pass filter is 

 greater than 



f R(co)dc 



The function R(co) represents, as above, the statistical energy spectrum 

 of the interference. 



Comparison of these formulas shows at once that, unless the energy 

 of the random interference is largely confined in the range co<a> £ , 

 little protection is afforded by the high pass filter. 



Appendix I 

 Derivation of Wave-Filter Indicial Admittances 



1. Low Pass Wave- Filter, Type Lid. 



The derivation of the indicial admittance of this type of filter is 

 given in detail by one of the writers in a previous paper. 25 The 

 method of solution there employed, which is quite generally applic- 

 able to periodic structures, consists in writing down the Heaviside 

 Expansion formula for the current in the wth section of a filter of s 

 sections in length (s>n), short circuited at the 5th section. The 

 expansion is converted into a definite integral by letting 5 become 

 infinite and the formula becomes that of the indicial admittance of 

 the nth section of an infinitely long filter. For the non-dissipative 

 filter having mid-series termination, this procedure leads to the 

 formula 



A n {t) = j- I dx\ - / cos (2ttX)« cos (xi sin \)d\, x = o> c t, 



k JO 7T Jq 



which is identifiable, from known formulas, as 



A n (f)~^f J 2 n(Xi)d Xl . (1.1) 



A much more direct and flexible method of solution and one which 

 avoids the necessity of setting up the Heaviside expansion formula 



25 Transient Oscillations, Trans. A. I. E. E., 1919. This paper should be con- 

 sulted for the details of this method. 



