TRANSIENT OSCILLATIONS IN ELECTRIC WAVE-FILTERS 43 



where the primes denote differentiation with respect to the argument 

 z. The solution of these differential equations is based on the ap- 

 proximation, valid only when z>n, 



2^ J n (s) = - q% J n (s) , q„= Vl-n 2 /z*. 



To this approximation, which becomes more and more accurate as 

 % and n increase, the differential equations are satisfied by solutions 

 of the form 



and 



S n =-^—. ■, J„(z)+A sin (vz-a), 



C„=— 2 :, J' n (z)+A cos(rz-a). 



A and a in the complementary terms are arbitrary constants, which 

 must be determined. These complementary terms, periodic in vz, 

 are evidently the ultimate values of the integrals when z approaches 

 infinity, which are known. Other considerations, however, show that 

 these terms should be omitted when v<l and z<n/\/l — v 2 . Conse- 

 sequently we arrive at the following approximations. 31 



For v<\ and n<z<n/vl — v 2 > 



* Qn 



and 



q„=\/l—n 2 /z 2 . 



This approximation is not accurate at z = n, and breaks down at the 

 critical point z = n/v 1 — v 2 . In the interval between, however, it 

 is a fair approximation, particularly when v is nearly equal to unity 

 and n is not too small. 



For v<l and z>n/v 1 — v 2 , 



S n {z\v) = - i jJ n {z) + , ■ sin (vz — n sin -1 ^, 



v ~ Qn yl-i' 2 



and 



C„(z\p) =- :,/«( z )+ , cos (vz-n sin - ^)- 



" -Q'» Vl-c 2 



31 The qualitative properties of these definite integrals can be deduced from th e 

 principle of stationary phase (See Theory of Bessel Functions, G. N. Watson, p. 229) , 



