TRANSIENT OSCILLATIONS IN ELECTRIC WAVE-FILTERS 35 



It is now convenient to introduce the symbolic notation 

 and 



P H = COS [X ( VWtf 2 - 1) 1 J*n(y) 



<2„ = sin [x(VT=7d*-l)]J 2n (y) 



(3.12) 

 (3.13) 



where the symbol d denotes the differential operator d/dy operating 

 on J2 n {y). The actual numerical significance of these formulas is 

 gotten by expanding as in (3.11). 



With the same symbolic notation we get finally, 



^„ W =^sin(*VTV^^==/ s ,( y ), (3.14) 



The exact solution (3.14) is too complicated, as it stands, to be of 

 any practical value. Fortunately, however, it is possible to sum the 

 expression asymptotically, and the resultant formula shows clearly the 

 behavior of A n {t) and in particular the character and magnitude of the 

 errors in the approximate formula of the text. 



When y is large compared with (4w) 2 , 



Jiniy) = 



ttv 



cos ( y 



4rc + l 



') 



and 



£/*<»'* (-D'>| 



2_ 



7ry 



b- 



3 2^(25-1) 

 2 4/ 



/ 4n + l 

 cos^y -— 



(3.15) 







s . ( 4n-f-l 



-- sin I y -. — 



y V 4 



-) 



to order 1/y 2 . 



If this expression is substituted in the expanded form of (3.14), 

 some rather intricate and tedious operations finally give as the asymp- 

 totic limit of A n (t) 



w m k \ ixy 



(l- -p 2 + . . Jsin(x\/l+P 2 )cosU--^ — tt) 

 -\2 P +' •) cos C 1c V / l+P 2 )sin^--^ — ttJ 



(3.16) 



The coefficients of the two terms of (3.16) are even and odd power 

 series in p respectively, powers of p beyond the second being neglected. 

 Formula (3.16) is important, as showing the effect of the band 

 width, that is of the parameter p, on the indicial admittance. It can 

 be used for numerical computation, however, only when ;y>(4n) 2 . 

 A corresponding formula, valid over a much wider range, is obtain- 



