KENNELLY. — BUILDING UP VOLTAGE AND CURRENT. 707 



The greater a,, the shorter the wave-length, or the more waves that can 

 be filled into the diagram. 



In the case represented by Figure 2, a = a^ -\- jno = 0.07675 + 

 j 0.7854 per kilometer. After running, say thirty kilometers, the wave 

 will have dwindled by the atteiiuation-coefhcient e-soo.orers+jojsM) _ 



^-(2..XK6 + i 2^^.562) ^ ^-2.3026 ^ ^-JZrm_ r|,J^g ^^^^ f^^^^j, -^ g^^^J ^^ nUmcric 



0.1. The second is equal to a negative angle of 23.562 radians, or 3| 

 complete rotations or cycles. The waves of current and voltage on 

 arriving at B have thus each dwindled to 10 per cent of their original 

 amplitude, and have also fallen 3f cycles behind the phase existing 

 at A. The phase of the wave remains the same as it advances, a 

 crest remaining a crest, but the phase at A is constantly advancing at 

 the angular velocity w radians per second, so that with respect to the 

 phase at A, that of the advancing wave is constantly falling behind. 



After advancing another thirty kilometers, or to C, Figure 2, sixty 

 kilometers from A, the voltage and current waves will again shrink to 

 10 per cent of their amplitudes at B, or to 1 per cent of their original 

 amplitudes, and will have fallen 7^ cycles in phase behind the voltage 

 and current phases at A, respectively, the attenuation-coefficient hav- 

 ing become e-'-'®'^2 + -''^^'-'" = 0.01 \270O°. The original phase differ- 

 ence between the voltage and current waves at A is equal to the angle 

 of the vector z^, the initial sending-end impedance (17). With negli- 

 gible resistance and leakage conductance, formula (17) shows that the 

 angle of Zq would approach zero ; so that on such a circuit the voltage 

 and current would be in phase at the start, and also at any point along 

 the line. Whatever the original phase difference might be, according 

 to (17), it would be maintained throughout the first run along the 

 line. 



If we represent at E, Figure 3, a unit vector rotating in a vertical 

 plane about a horizontal axis 0, with the angular velocity oj radians per 

 second, its projection on the horizontal plane 6 2 will at any instant 

 represent the actual voltage applied to the end A of the line. At 

 a distance of one kilometer from A, the phase of the outgoing wave 

 stream will be 1, or 45° behind E. The amplitude will have shrunk 

 to 1' or 0.923, an attenuation of 7.7 per cent. The projection of 

 1' on the horizontal plane 6 2 will give the amplitude of the wave 

 at the same instant. At a distance of another kilometer beyond, or 

 two kilometers from the origin, the phase of the wave stream will be 

 90° behind E, and the amplitude will have again lost 7.7 per cent, or 

 will have fallen to 0.853, 85.3 per cent of the original. By continuing 

 the examination, we should find that at the instant considered, all 

 points along the line would have a wave amplitude determined by the 



