12G2 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 



For mathematical solution, (9), (10) and (11) may be put in the fol- 

 lowing form: 



^1 + aPi = (12) 



dz 



^ + /3Px - auP^ - auPn = (13) 



dz 



dP 



^ + aP„ - a.iP. = (14) 



dz 



in which a = aih -\- dix 



13 = a^h + ctxt 



The general solution for (12), (13) and (14) is given by the following 



Pi = ki e~"' (15) 



p, = fc2 8^^' + ks e'" (16) 



P„ = -k,e-"' + ^ (ri + ^)i'' + - (r2 + ^y (17) 



where 



n _ -(« + ^) ± V(« - iS)- + 4ai,a,i 



^2 



(18) 



The positive sign is to be associated with ri and the negative sign with 

 r2 . For the boundary conditions, at 2 = 0, 



Pi = Po 

 P. = 0, 



Pn = 0, 



that is to say, the input to the transmission medium being zero in both 

 the X mode and reconverted energy mode, then the solution takes the 

 following form: 



Pi = Poe""' (19) 



p _ dlxPo 7- 12 (llxPo rzz /()f)\ 



(r-i - ra) (n - r^ 



