

AVAVEGUIDE MODES IX GYROMAGXETIC MEDIA 



415 



sponding to some linear combination of the positive and negative values 

 of kz for one of the values of kz-. Examination of (9) reveals that the 

 ratio of Ey to E^ , the field components tangent to the bounding walls, 

 to be independent of the sign of kz . Hence, each partial wave has an 

 individual value of this ratio irrespective of its standing wave distri- 

 bution in the z direction. It is thus impossible, in general, to provide 

 a mutual cancellation of two or more partial waves at the electric 

 walls by combinations of such partial waves, with the consequence that 

 each partial wave must individually satisfy the boundary reciuirement. 

 We find, then, that each partial wave takes on the familiar condition 

 kz = mir/h. 



The parallel plane wa\'es now will be appropriately oriented and 

 superposed to satisfy the side wall boundary conditions in the rectangular 

 guide. Since, as shown in Fig. 2, mutual cancellation is required on the 

 side walls of the rectangular guide, the rate of vertical variation must 

 be identical for all the component parallel plane waves; thus 7n is a 

 constant of the waveguide mode and k, is uniquely specified. 



Two essential characteristics thus define a rectangular waveguide mode 

 in a transversely magnetized, ferrite filled, medium. 



1. The modes are ordered by integral values of m in the relationship 

 /.% = 7mr/b. 



2. The propagation constant ky is uniquely specified. 



Standing waves ma}' now be formed in the z direction satisfying electric 

 boundaiy conditions at the parallel planes. Each partial wave of the 

 electric field may then be expressed as follows corresponding to its 

 appropriate value of p: 



E = 



viir 



biCE 



H. 



I . 1 (mir\ \ . niTT 



I sm 



VlTZ 



— i(mx/6) (1— p/p) iy 



(10) 



Let us now specialize our analysis to the small guide case. The require- 

 ment of birefringence to produce small guide propagation demands 

 that A-2 be non-\-anishing and that 7n take on an integral value of unity 

 or greater. "We have, from (7), the two limiting values of p corresponding 



