ELECTRIC CIRCUIT THEORY 73 



to secure the condition R/L = G/C, the line is thereby rendered dis- 

 tortionless but the attenuation is doubled. 



One of Heaviside's most important contributions to wire trans- 

 mission theory was to point out the properties of the distortionless 

 line, its approximately realizable character, and to base on it a correct 

 theory of telephonic transmission. 



The character of the wave propagation when the parameters 

 p and (T are not restricted to special values, can only be roughly in- 

 ferred from inspection of the formulas, and then only when the prop- 

 erties of the Bessel function lo and Ji have been studied. Fortunately 

 these functions have been computed and tabulated for small values 

 of the argument, and have simple asymptotic expansions for large 

 values. It is therefore a simple matter to compute and graph a 

 representative set of curves which show the current and voltage 

 waves for various values of p, a and x. For this purpose it is con- 

 venient to introduce a change of variables and write: 



T=Vt 



a = p/v 

 b = <T/v 

 whence the formulas for current and voltage become: 



L _ (210a) 



^^,-a.+,.r ^-^^(Hg^) ,,. (211a) 



Figs. (9) to (18) give a representative set of curves illustrating 

 the form of the propagated current and voltage waves for different 

 lengths of line, and different values of the line parameters a and h, 

 or p and a. 



The curves of Figs. (9) and (10) show the current entering the 

 line in response to a unit e.m.f. applied at time / = 0. The line is 

 assumed to be non-leaky (& = 0) and is computed for two different 

 values of the parameter a. We see that the current instantly jumps to 

 the value \/C/L and then begins to die away, the rate at which Jt dies 



away depending on and increasing with the parameter ^~ 'o 'VY' 



If we now consider a point x out on the line, the current is zero 

 until T = x, at which time it jumps to the value \/C/L e"*"". It then 



= V 



