RECTANGULAR WAVE GUIDES U'> 



in which w = 2, 3, 4, • • • and tlic primes on /•',„ and /I have the same 

 significance as in (1-4) and (1-5). 



The principal problem here is to solve equations (2-4) and (2-5) simul- 

 taneously subject to 



(2-6) 

 Fi = Tf,(T), v-^ -\-x 



which again corresponds to a unit wave in the dominant mode incident from 

 the left. Tn(v) and the remaining F's correspond to outward traveling 

 waves as before. F-i , F3 , ■ ■ ■ aW approach zero as z; — > — =0 . 



3. Remarks tvi Solving the Equalioiis of Sections I and 1 for the Reflection 

 Coefficient 



Suppose that we have a system in which the wave propagation is governed 

 by the single differential equation 



—^ - h'y = (3-1) 



av~ 



where // = h(v) is a positive imaginary function of v, twice differentiable and 

 such that h -^ ic, c being a constant; as r ^ — x . We desire the solution 

 of (3-1) which, together with its first derivative, is continuous everywhere 

 and at ± X satisfies the conditions 



y = g-icv _^ J^gicv^ ^, ^ _ oo (3-2) 



y' + (// + h'/{2h))y -^ 0, r -> ^ {i-i) 



The constant R (the reflection coefficient) is to be determined. Condition 

 {i-i), in which the primes denote differentiation with respect to v, is sug- 

 gested by the fact that we want y to represent a wave traveling in the positive 

 V direction (the factor exp (/oj/) is suppressed). In writing ii-i) we have 

 assumed that // is such that for large values of v the two solutions of (3-1) 

 are asymptotically proportional to* 



y = h-'e"-", (3-4) 



^ ^ ^(r) = icv + [ {h - ic) dv. (3-5) 



Physical considerations suggest that solutions satisfying (3-2) and {^-i) 

 exist in most cases of practical importance. However, if the function h is 

 picked arbitrarily the corresj)onding solutions may be incapable of satisfying 



* S. A. SchelkunotT- mentions that this approximation, sometimes designated by 

 "WKB", goes back to Liouville. The ideas we shall use are quite similar to those in 

 SchelkunotT's paper. 



