Damped Electrical Oscillations along Parallel Wires. 299 

 ation in their passage along the leads we should have 



j3z-gt = when z= P - 3 i. e. /3= ^. 



2 a, p 



In general, it is plain that ( ft ) measures the attenuation. 



Inserting the complex variables in equation (-1; we have 

 v z (ft-ia) 2 =:{p-q^ip){cr — q + ip); 

 ... v ^- a * )=( f-f- q[p + (T)+p(7 . .... (8) 



and 



2v 2 aft-=p{2q-p-a); (9) 



whence 



vW + **)= >/{p* + (q-p)*\{p a +{q-<rr\. . (10) 



Velocity of Propagation and Attenuation. — In actual cases 

 p and cr are small compared with p. If the damping is con- 

 siderable, q may be comparable with p. Accordingly we 

 expand the right-hand side of (10) in ascending powers of p 

 and a and solve for va and vft. As far as terms of the third 

 order in p and cr we find • 



^f^ + a ^g#+ cd 



Hence the velocity of propagation 

 and the attenuation 



= <7" o- 1 P +Q '-i- y^" 0- ) 8 + (% 2 -y)(p+^)(p-Q-) 2 , i m 



If p = <7, there is no distortion, the velocity of propagation 



is v, and the attenuation is ^— or -~- for all frequencies : and 

 7 Ly bv l 



the damping has no effect on these quantities. 



We have an interesting particular case when p = cr = q. 



Then /3 = 0, and the state of affairs is given by 



Y =Y e-* t sin (pt— ~\ 



Here the damping and the attenuation are balanced, so that 



