536 BELL SYSTEM TECHNICAL JOURNAL 



Consider a 1000-cycle sine wave which is modulated by a 25-cycle 

 sine wave in such a manner that the envelope just reaches zero once 

 per cycle of the modulating wave. This wave is found on analysis 

 to consist of three components, namely, 1000 cycles of two units 

 amplitude, 975 cycles of one unit amplitude, and 1025 cycles of one 

 unit amplitude. At the start, it is somewhat simpler to consider this 

 case with the 1000-cycle component removed. In other words, the 

 current transmitted through the system now consists of 975 and 1025 

 cycles in equal amounts. This value at the sending end may be 

 conveniently written 



sin 975 2^ + sin 1025 2^t. 

 The equivalent graphical expression is 



2 cos 25 lirt sin 1000 lirt. 



Now suppose that the 975-cycle current suffers a phase change of 

 (3975 during transmission and that the 1025-cycle current suffers a phase 

 change of /3io25, then the analytical expression for the current at the 

 receiving end is 



sin (975 livt - ^^n) + sin (1025 27r/ - ^xo2h). 

 The corresponding graphical expression is 



n I oe T^ /3i025 "~ ^975 \ . / inAA^T~^ ^1025 + /3975 



2 cos 25 27r/ ^ sm 1000 27r/ ^^ 



In comparing the graphical expressions for the current at the sending 

 end and the current at the receiving end, it is apparent that the only 

 changes that have taken place are phase shifts of the 1000-cycle carrier 

 wave and of the 25-cycle modulating wave. The phase shift of the 

 25-cycle modulating wave represents the actual delay of the deforma- 

 tion of the carrier wave. If the circuit is sufficiently long so that this 

 phase shift amounts to one complete cycle, then the corresponding 

 delay equals one period. For any other delay, the phase shift and de- 

 lay are, of course, proportional. It will be apparent, therefore, that 

 the delay may be represented by the following equation: 



^ 2 X 25 (27r) Aoj ' 



where 7a is expressed in seconds, the numerator in radians and the 

 denominator in radians per second. 7 a, the value of the delay of this 

 envelope, is according to our previous definition substantially the 

 envelope delay. 



