RECTANGULAR WAVE GUIDES 145 



Because of the second of equations (3-20), at least one of the ^„/s becomes 

 infinite as tpt passes through zero. Hence we expect the ap{)roximate 

 solutions to be valid only over intervals in which none of the c^^'s become 

 zero. When we hold fast to a given solution yi , y2 , • ■ • Jn oi (3-18) as v 

 j)asses through a value which makes one of the (pi's zero, we expect the set 

 of constants d( to be replaced by a new set (Stokes phenomenon). What 

 is the relation between the new set and the old set? It may possibly be 

 much the same as for the case .V = 1 (see JeflFreys^", Langer^ and Furry'^). 



4. When the matrix A is such that the approximation (3-27) remains 

 valid over the entire interval — oo < ^; < qo (unfortunately this restriction 

 prevents us from applying the following results to the horn of Section 4) 

 the matrix analogue of (3-15) is the integral equation 



y{vo) = S^e-^'J^ -f \S, \ e---' [S' + 2^S' -f ^S'\y{v) dv (3-29) 



J— 00 



+ \So / e^'~^ [S' - 2^S' - ^S'\y{v) dv 



where the subscript zero on ^ and H indicates that they are to be evaluated 

 at D = I'o . The column matrix y{v) giving the solution of (3-29) is that 

 solution of y = Ay which satisfies the conditions 



y = Se~J+ -f Se-f-, zj -> - oo (3-30) 



y = Se~"g+, z) -^ oo (3-31) 



where the column matrix /+ (corresponding to the incident wave) is given 

 and/~ (corresponding to reflected wave) and g+ (corresponding to the trans- 

 mitted wave) are to be determined. The elements of /+, /~, and g+ are 

 constants. It is further supposed that S satisfies the conditions 



S'S - S'S = 0, z)^±oo 



which are certainly met if the elements of A approach constants at ± «> . 

 In the wave guide problem we assume <pt to approach the limit 5^ as v 

 approaches — qo . For this case it is convenient to deline the ^th element 

 in the diagonal matrix S as 



^( =^ 8iv -\- I {(ft - 5^) dv 



J— 00 



In any event we have 





^ dv 



