112 BELL SYSTEM TECHNICAL JOURNAL 



minus sign when Re is given. These relations may be proved by substituting 

 various solutions of equation (2-3) for Q and Q in the equation 



^ dv ^ dv 



Qf-Qf] (3-9) 



where I'l and V2 are large enough (t'l negative and v^ positive) to ensure that 

 Q and Q have reduced to exponential functions of v. Equation (3-9) follows 

 from Green's theorem. When Q is taken to be the solution for which (2-7) 

 and (2-8) holds and Q its conjugate complex Q*, equation (3-8) is obtained. 

 Keeping the same solution for Q but now letting Q denote the solution 

 corresponding to an incident wave of unit amplitude coming in from the 

 right : 



Qi = Tie'''" , v-^ —CO 



gives T = Ti where we have dropped the subscript E and have assumed 

 that g{v, 6) may be unsymmetrical. Taking Q to be Qi gives 



RTt + RtT = 



which is the relation sought. In the symmetrical case R — Ri, R/T + R*/T* 

 is zero and hence R/T is purely imaginary as was mentioned above. The 

 same relations hold for R„ and Th ■ These results are special cases of a 

 more general result which states that the "scattering matrix" is symmetrical 

 and unitary for a lossless junction.^" 



4. Approximate Solution of Integral Equation 



A first approximation to the solution of the integral equation (3-5) is 

 obtained when we assume that the non-uniformity of the propagation 

 constant has no efifect on Q. Thus we put 



Q^'\v, 9) = e-"-" (4-1) 



in the integral on the right and obtain an expression for the second approxi- 

 mation Q'-'^v, d), and so on. Here we shall not go beyond Q^-^(v, 6). 

 It is convenient to expand g(v, 6) in a Fourier cosine series 



00 



g(^', ^) = S «n(t') cos nd 



(4-2) 

 an(v) = -" / g('', 0) cos nd dd, eo = 1; e„ = 2, n > 0. 



TT Jo 



