106 BELL SYSTEM TECHNICAL JOURNAL 



values at minus infinity and the same values at plus infinity. WTien this 

 assumption is not met, a conformal transformation may still be used to 

 carry the system into a straight guide. However, there appears to be some 

 doubt as to the best way of dealing with the resulting partial differential 

 equation. One method, discussed in the companion paper , leads to an 

 infinite set of ordinar>' linear differential equations of the second order. 

 Again, possibly the Green's functions appearing in Sections 3 and 5 may be 

 replaced by suitable approximations. 



/. Representation oj Field for Corner or Bend in Rectangular Guide 



Quite often waves in rectangular wave guides are classed as "transverse 

 electric" or "transverse magnetic". However, for our purposes it is 

 more convenient to class them as "electrically oriented" or "magnetically 

 oriented" waves.*'' Thus, the electric and magnetic intensities are obtained 



(1-1) 



1 a'^ „ - „ , 1 d^B 



£{. = - ico^xA + — rrr: H^ ^ - iweB + . 



' toeaf' 7WMar- 



by e'"' and taking the real part. Here w, p, and e are the radian frequency, 

 the permeabiUty of the medium filling the guide (m = 1.257 X 10~* henries 

 per meter for air), and the dielectric constant of the same (e = 8.854 X 10""'- 

 farads per meter for air), respectively, .t, y, and <: constitute a right-handed 

 set of rectangular coordinates in which the f axis is normal to the plane of 

 the bend. Equations (1-1) may be verified by substituting them in 

 Maxwell's equations. 



The potentials A and B satisfy the wave equation 



d'^A , d-A , d'A 2 , 



+ + TT — (^ A 



dx^ ^ df ar (1.2) 



a = tcoViue = i2ir/Xo 



where Xo is the wave length in free space corresponding to the radian 

 frequency w. 



WTien the electric vector lies in the plane of the bend, as shown in Fig. 1, 

 and the incident wave contains only the dominant mode we set 



^ = 0, B = Qsin (ir^/a) (1-3) 



