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BELL SYSTEM TECHNICAL JOURNAL 



able from the expression derived in Appendix II for the Bessel func- 

 tion, namely 



Jzn(y)=B% n {y) COS &2n(y)- 



If this expression is employed instead of (3.15), we get corresponding 

 to (3.16), 



A n (t) = — h B 2n (y)l 



w 



(l-|a 2 + . . .) sin (.r\/l + <T 2 )cosfi 2M (y) 

 — (s ff + • • • ) cos (xVl + c 2 ) sin ft 2n (y) 



(l-^ 2 + • . •) sin (xVi^rV) Jin (y) 



+ (|p+ ,-"-) COS (xVl+O Jin (y), 



(3.17) 



(3.18) 



where (T = pq 2n = p\/\ — (2n/y) 2 . 



Formula (3.17) is valid when y>2n, and ultimately approaches the 

 limit (3.16) as y becomes indefinitely large. 



We are now prepared to discuss the character of the approximations 

 of the formula of the text, which may be written as 



w 



2i>i m k 



Bin (y) j sin [x + ft 2 „ 60] + sin [x - ®2n(y) ] 



(3.19) 



Correspondingly (3.17) may be written as 



2to m & 



B in 6) 



+ (l+^ + . . .)sin[xVl + ^ 2 -Q 2M 6)]. 



(3.20) 



Comparison of (3.19) and (3.20) shows that the approximate formula 

 of the text ignores slowly variable correction factors in the ampli- 

 tudes of the component oscillations, and a slowly variable change in 

 their frequencies. For band pass filters employed in practice these 

 corrections are not only slowly variable but in most cases are quite 

 small. In any case, it is important to observe that failure to include 

 these corrections does not appreciably affect any essential features of 

 the building-up phenomena discussed in the text. Consequently the 

 deductions from the formula of the text are valid not only for narrow- 

 band pass filters, but also for filters of quite wide bands. This state- 

 ment is substantiated by the fact that the steady-state characteristics, 



