620 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



replaced by 



^ + ^ (? = 0, r = L, r = a (C7) 



dr r^ 



Equation (C6) can be solved immediately to give 



S,{t) = e-'''">' (C8) I 



and if we assign the subscript 77 to the G going with r? the most general 

 solution of (CI) and (C2) will be 



P = Z A,Gr,(r)e-"'''°' (C9) 



where the A,, are arbitrary constants so determined that (C3) is satisfied. 



Equation (C9) shows that in reality there exists, for this problem, a 

 spectrum of relaxation times, I/t^'Do . After a brief transient period many 

 of the higher order terms will decay away and eventually only the first 

 two terms will have to be considered. Finally when equilibrium is at- 

 tained only the first term Avill survive. 



The last statement implies that 77 = 0, is an allowable eigenvalue, i.e., 

 that the first term is independent of time. That this is so can be proved 

 by solving (C5) for 17 = 0, and substituting the result in (C7). Thus 



Go(r) = exp (f^^ (CIO) 



and this does satisfy (C7). p can then be approximated after the transient 



by 



p = Ao exp (j^ + ^1 Gi(r)e-''i-^'" (Cll) 



from which it is obvious that the relaxation time dealt with in section X 

 is 



T = -hr (C12) 



In principle it should be possible to evaluate Gi by the straightforward 

 solution of (C5) and determination of the second eigenvalue through 

 substitution of this solution in (C7). In fact this represents a rather un- 

 pleasant task since G is a confluent hypergeometric function. Therefore 

 we shall follow an alternative route based on the assumption that by the 

 time (Cll) applies the flux 4xr"J*(r), where J* is given by (10. IG), is 

 almost independent of r. The reader is referred to some related papers ' 



