wliei 



Transmission of Electric Waves along Wires. 



A COS a. = — 4. — ' £ 4_ = 



A' 



3t33 



A sin a = — "2^/A 1 



— ,,i 



Bcos/3: 



^?#-*- 



A' 



= A = 1 + 



Bsin/3= -2 x '/&' = V = 



X ! = LvS'p. 



We have then finally 



(8) 



A=4-v / a a + a' 2 , tan a =a f ]a, 



' \ . . (9) 



B = + V^ 2 + &' a and tan {3=b'/b. 



The algebraic signs of a, a 1 , b, and b' being preserved show in. 

 which quadrants a and /3 should be taken. 



17. We see from (6) and (7) that the reflected and trans- 

 mitted wave-trains consist each of a damped periodic part and 

 a purely exponential part. The exponential part we shall 

 have to omit from the first portion of the subsequent analysis, 

 because we have no experimental proof of the exact shape of 

 the head of the incident wave-train. For the same reason 

 we shall at first omit the i also. The greatest correction 

 which this approximate theory can in any possible case 

 require will be subsequently discussed and evaluated. (bee 

 arts. 60 and 61.) 



18. Under these simplifications we may compactly repre- 

 sent the three wave-trains as follows : — 



Incident wave-train, real part of ei*— ^')?^, 



Reflected wave-train, real part of (a + a'i)^*'— A )JP', 



Transmitted wave-train, real part of (b + b'^ett-fypt; 



where a, a' ', b and V have the values assigned to them in (8) 

 and have been confirmed by Heaviside's operational methods. 



19. It may be noted in passing that if the incident wave- 

 train were not damped, the reflected and transmitted trains 



7T 



would differ in phase by -r exactly. For on putting k — {) 

 in the expressions for tan a and tan /3 they become 



1 



M 10 ) 



—2 



-x' 



- 7 and — ^respectively, hence their product is — 1. This 



