Intelligence and Miscellaneous Articles. 529 



The general solution of these equations is 



i = ^hke 



«£•* 



* .(t-Ckx) 



h, Jc 



where e is the Naperian base, and h and h are constants to be 

 determined. 



If the impressed E.M.F. is harmonic, and at the origin 

 £=Esinw<, where 10 denotes angular velocity, the solution for the 

 potential at any point of the conductor at any time becomes 



e=Ee ±paf sin («*+«*?) ( 3 ) 



The solution for the current at any time across any section of the 

 conductor is 



. -En/Cw 



€±px sin^±a.r+tan-i|). . . . (4) 



>/Im. 



In these equations Im. denotes the impedance, (E 2 +LVf ; 



/<5Z /C^ - 



P == \/ 2" Vim.— Lw; and a=\/ -~- \/Im. -f- Lo>. 



The solutions in equations (3) and (4) show that the potential 

 and current are propagated in harmonic waves whose amplitudes 

 decrease with the distance from the origin according to a loga- 

 rithmic decrement. At any point of the conductor the potential 

 and current vary as simple harmonic functions of the time with 

 constant amplitudes which are different for every point of the 

 conductor. The current wave is propagated in advance of the 



potential wave by an angle such that tan 0= -. This phase- 

 difference diminishes with increase of frequency when there is self- 

 induction, but becomes a constant angle of 45° when L = 0. The wave- 

 length is — and the rate of propagation is — . The wave-length 



and rate of propagation each become less as the self-induction 

 increases. The wave of higher frequency will have the shorter 

 length and be propagated the faster. This difference in rate of 

 propagation of waves of different frequencies is most marked when 

 there is no self-induction. 



The distance at which the amplitude decreases to — th of its 



value is — =^— ~ — r; the time for the decrease is — — :> The 

 p 2tt tan d o) tan 



rate of decay is most rapid when there is no self-induction. The 



waves of higher frequency decay more rapidly than those of low r er 



frequency ; when there is no self-induction this difference in the 



rate of decay is the greatest. 



