IRREGULARITIES IN WIRE TRANSMISSION CIRCUITS 557 



The echo current which is joined to the unit received wave affects 

 the final resultant and therefore the effective loss of the line. From 

 equation (28), neglecting higher-order terms, the attenuation of the 

 whole line is increased (in nepers) by 



e + 



(f)^ \ nb"^ 



(35) 



The standard deviation of the attenuation (A, in nepers), from equa- 

 tion (34) and neglecting higher order terms, is 



a/(A - A)2 = 



"sTT 



(36) 



Appendix III 



Sinuosity 



The following is a derivation of the sinuosity of the attenuation, 

 defined as the standard deviation of the difference A(/ -f A/) — A(/). 

 Here A(/) is the loss in the circuit at the frequency, /. 



For practical purposes, the difference of the expression Ecr — Ecr 

 at two discrete frequencies is 



d(Ecr — Ecr) 



X = 



df 



A/, 



(1) 



whose standard deviation will be derived below. From values of Ec 

 and Ecr given in Appendix II we obtain 



J-'CT J-'C 



3=0 



Y, UmB"' cos mcf) 



j-.d{B COS0) " 



+ [_! - B cos 4>']n¥, (2) 

 diB™' cos m4>) 



df 



A/ 



, d cos(f) .\dB] 



n¥ { B —^ h (cos 0) -jj } 



, y-^ rr \ T, d COS md) , , , dB^ 



+ H Hm{ B^ + (cos m<i>) -jj 



A/ 



nh^{BQ s'm (f) — D cos 0) + 2] niH„, 



X (B^'Q sin mcf) - B"'-^ cos w0) 



A/, (3) 



