CON FORMAL TRANSFORMATION 111 



where 7m is given by (2-1). The term e~'*''o comes from the first integral 

 on the left side of (3-4) as vi — >• — oo . Equation (3-5) is a general equation 

 which may be applied to a number of wave guide problems by choosing a 

 suitable function g{v, 6). For the corner of Fig. 1 g{v, 6) is given by (2-6). 

 If g{v, 6) approaches zero when | v \ becomes large, as it does for the 

 corner, expressions for the reflection coefficient Rg and the amplitude Te 

 of the transmitted wave may be obtained by letting I'o -^ ±20 in (3-5). 

 For ver>' large values of | r© | the contributions of all the terms in the summa- 

 tion e.xcept the first (w = 0) vanish. Comparison of the resulting expression 

 for Q(vo , do) with the limiting forms (2-7) and (2-8) defining Re and Tb gives 



Re= -^ r dv f dd g(v, e) Q(i, d)e-''' (3-6) 



Zir J-x, Jo 



T,= \-^ r dvf dd g(v, d) Q(i, 0)6'" (3-7) 



Zir J-to Jo 



Since the integrands involve the as yet unknown Q(v, 6) these expressions 

 are not immediately applicable. In fact, if we knew Q{v, 6) it would not be 

 necessary to use these integrals for Re and Te — we could simply let i; — > ± oc 

 and use (2-7) and (2-8). Nevertheless, (3-6) and (3-7) are useful in obtain- 

 ing approximations to Rb and Te when approximations to Q are known. 



In Appendix IV it is shown that Rb is the stationary' value, with respect to 

 variations of the function Q, of an expression made up of integrals containing 

 Q in their integrands. From the integral equation it follows that when 

 k-^Q, i.e., when the frequency decreases toward the cut-off frequency of the 

 dominant mode, Q becomes approximately exp {—ikv). Furthermore, Rg 

 approaches zero. This is in contrast to the apparent behavior of Rh which, 

 according to the discussion given in Section 5, may possibly approach — 1 

 under the same circumstances. Thus reflections from the two types of 

 corners, or more generally, irregularities in the E plane and in the H plane, 

 appear to behave quite differently as the cut-off frequency is approached. 



Rb and Tb are not independent. Since the energy in the incident wave is 

 equal to the sum of the energies in the reflected and transmitted waves we 

 expect 



ReR*e + TeT*e - 1, (3-8) 



where the asterisk denotes the conjugate complex quantity. In addition, 

 there is a relation between Rb and Tb which for a symmetrical irregularity, 

 i.e. for g{v, 6) an even function of v, states that the phase of Re differs from 

 that Te by ±7r/2. In this special case Tb is determined to within a plus or 



