298 Mr. W. B. Morton on the Propagation of 



which can be made to represent damped periodic vibrations 

 by giving complex values to m and n. 

 The equation (3) now becomes 



mV=(/D + ra)(o- + 7i)s (4) 



and the connexion between C and V is given by (1), viz. 



L -R^Ln ^ 



To find the effect of a pure resistance B/ between the ends 

 of the wires, as in Dr. Barton's experiments, put Y 1? Y 2 , 

 G 1} G 2 for the potentials and currents in the incident and 

 reflected waves respectively. Then we have 



p _ mY-i n _ -mVo 



B + W z R+hn 3 



also the total potential-difference V x + Y 2 is connected with 

 total current G 1 -i-G 2 through resistance B' by Ohm's law, 



Vj+V^B^Ci + Co). 



These equations give for the reflexion factor 



Vj _ B + L?z-7»B / 



Y x B + Lw + mB/ (6; 



If the circuit be distortionless and B/ = Lv, then, as 

 Mr. Heaviside showed, the absorption of the waves by the 

 terminal resistance will be complete. We may regard this 

 as the critical resistance for the circuit, and we shall express 

 B' in terms of it by putting B'^^Lv. We then have 



Y 2 _ _ p + n — mvx 

 Y l p + n + mvx 



Damped Wave-Train. — To pass to the case of a damped 

 train transmitted from the origin in the positive direction of z 

 we put 



in=—j3 + ia } n=—q + ip. 



The difference of potential between the wires at any point 

 after the head of the wave-train has reached this point is then 

 represented by an expression of the form 



Vo^-a'sin {pt—az). 



The velocity of propagation is -, the frequency -£—, the 



2-7777 a 



logarithmic decrement — -. If the waves suffered no attenu- 



P 



