52 BELL SYSTEM TECHNICAL JOURNAL 



Comparison with the operational equation shows that they are iden- 

 tical, within a constant factor provided we put X = q;/4. Conse- 

 quently the solution of (165) is 



'=4^m'-''"-^ 



-^ e-i/T 

 TrRr 



which agrees with (168). This, it may be remarked, is an excellent 

 example of the utility of the table of integrals in solving operational 

 equations. 



This formula is easily calculated for large values of / by expanding 

 the exponential function; it is 



R^ VV;L^~i7/~^2!V7J -•••]• 



The propagation phenomena of the non-inductive cable are there- 

 fore determined by the pair of equations 



Vtti/o t\/t vxi/o 



(169) 



/= -7^- ^^ = A^^ e-''^ (170) 



, / 4f 



where r = 4//a= o>^7^. 

 x-KL 



Now an important feature of these formulas is that the voltage 



4 

 at point X is a function onlv of -itt.^^ /; that is, oi 4t divided bv the total 



x-RC 



resistafice and capacity of the cable from :\; = to x = x. The same 



statement holds for the form of the current wave: its magnitude, 



however, is inversely proportional to xR, or the total resistance of 



the cable up to point x. Consequently a single curve, with proper 



time scale serves to give the voltage wave at any point on the cable. 



Similarly a single curve, with proper time and amplitude scales, 



serves to depict the current wave at any distance from the cable 



terminals. These curves are given in Figs. 3 and 4. 



Referring to the curve depicting the current wave, we observe that 



it is finite for all values of />0; consequently, in the ideal cable, the 



velocity of propagation is infinite. This is a consequence, of course, 



of the fact that the distributed inductance of the cable is neglected. 



Actually, of course, the velocity of propagation cannot exceed the 



