REFLECTIONS FROM CIRCULAR BENDS 



307 



to the guide axis and, for the straight guide of the section, the guide walls 

 are sections of the planes x = 0, x = a, y = 0, j = ft. 



Thus, a general wave traveling in the positive z direction may be de- 

 scribed by the two functions (which represent the magnitudes of A and B) 



A = 2L ^in e~ "'"* sin {irmx/ a)cos {irny/b) 



(1.1-3) 



m 



1,2,3, 



B = Y. dt -^' 



» = 0,1,2,... 



cos {Trmx/a)sm (wny/b) 



m 



(1.1-4) 

 0, 1, 2, ... ; ;.= 1,2,3, ... 



where the coefficients gir» and d^n are constants and the plus signs indicate 

 propagation in the positive z direction. 



The propagation constant Tmn is obtained from 



(1.1-5) 



rL = ^2 + {irm/aY + {irn/by, a = i2ir/\o, 

 Xo = wavelength in free space. 



Equation (1.1-5) arises when the typical term in (1.1-3) is substituted for A 

 in the equation 



;i^ A ;i^ A A^ A 



(1.1-6) 



d'A 



a^4 ^^4 



dx' d'f- dz^ 



= a'A 



This and a similar equation for B are the forms assumed by (1.1-2) for the 

 rectangular coordinates of our straight guide. 

 The electric and magnetic intensities in the guide are given by 



(1.1-7) 



which follow from (1.1-1). 



It is seen that the wave is completely specified by the gin's and diiJs. 

 These may be arranged as (infinite) column matrices in any convenient 

 order. Thus in deahng with (1.1-3) and (1.1-4) we may write 



(1.1-8) 



