710 PROCEEDINGS OF THE AMERICAN ACADEMY. 



At any distance Li kilometers from A we should have 



If, however, the circuit be comparatively short, and be free, or opened, 

 at the distant end {Z — <^ in Figure 1), the wave stream on arriving 

 at B will be reflected back towards A. Let the length of the line be 

 L kilometers, then the voltage when it reaches B will have the value 

 Ei.-^"- and the current likewise /o (-^°-. The effects of reflection are 

 different on voltage and on current, i. e., on moving electrostatic flux 

 and moving magnetic flux, so that each must be considered separately. 



Considering first the voltage wave arriving at the distant open end 

 of the line, it rebounds therefrom at light-speed, or is reflected there. 

 The voltage in the reflected wave at B starts back at the same ampli- 

 tude and phase as when it strikes the free end. It is simply started 

 backwards along the circuit without undergoing any other change. 

 It returns towards A, attenuating as it goes, just as though nothing 

 had happened. At B, however, after the reflection, the voltage is 

 double that which occurred just before reflection ; for as soon as the 

 reflected wave starts back, there are the ^c--^" volts arriving at the 

 head of the incoming train, and also E^~^'^ volts starting at the head 

 of the returning train. The voltage at B was thus until the wave 

 arrived at B, then it was E^r-^"-, whereupon reflection immediately 

 occurred and made the voltage jump to 2 Ef~^"-; where a is a com- 

 plex number, the real part of 2 E^—^'^, being the maximum cyclic 

 value of the voltage at B, and 2 Ee-^'^^ sin {wt-a^) being the voltage 

 at any instant after arrival. 



By the time that the wave train gets back to A, the total distance 

 it will have travelled is 2 Z- kilometers, and the attenuation-coefficient 

 will be f.-^^'"-. The arriving voltage will therefore be Ef-"-^'"-. As 

 soon as the end A of the line is reached, the wave will go direct to 

 ground through the alternator. The e. m. f of the alternator does not 

 stop the wave, which goes straight through the machine (assumed as 

 impedanceless) without modification. A voltage wave passing through 

 a ground or short-circuit virtually swings round the goal and returns 

 on its path reversed. That is the same, however, as being reflected 

 with change of sign, or with a change of 180 degrees, or -k radians, in 



