REFLECTIONS FROM CIRCULAR BENDS 319 



Here [7] j represents the diagonal matrix havingyy the^**^ term in its principal 

 diagonal and 



T = k[yUk-' (2.1-11) 



Therefore we have shown that ^(2) is of the form (2.1-5) \\hich is what we 

 set out to do. 



It is rather difficult to compute T from (2.1-11) using only the above 

 definitions of k and 7,- for one must first obtain the functions (pj{x, y). In 

 deaUng with the rectangular guide it is easier to use equations (1.3-5) and 

 (1.3-13) to determine T. 



2.2 Reflection at a Single Junction 



Let a straight wave guide extending from 2=— ootos = Obe joined to a 

 curved guide of the same cross-section w^hich extends from s = to s = oc . 

 Let an incident wave 



00 

 ^i = Z hr.e-'''"'dr>Xx,y) (2.2-1) 



come in from the left along the straight guide. The hm's, are given constants, 

 the 5,„'s are the modal propagation constants for straight guides (for rec- 

 tangular guides they are the T^n's given by (1.1-5)), and dm{x, y) is the w*^ 

 eigenfunction for the straight guide (the product of a sine and a cosine for a 

 rectangular guide). 



WTiat are the reflected and transmitted waves set up by (2.2-1)? The 

 reflected wave is of the form 



00 

 ^r = Hf^e''^em(x,y) . (2.2-2) 



where the/^'s are to be determined. 

 If we assume the representation 



00 

 ^ = Z fJ^MSmix, y) (2.2-3) 



m=l 



to hold for all real values of z then, since <I> = <?>,• + <I>r for s < 0, equations 

 (2.2-1) and (2.2-2) show that 



Mn»(2) = h„.e-''- + fZ /'•", w = 1, 2, 3, . . . ; s < (2.2-4) 



Introducing the column matrices /i(z), h, f~ and the diagonal matrices 

 exp (±sro) where To is a diagonal matrix having bm as the w* term in its 

 principal diagonal enables us to write (2.2-4) as 



^{z) = e-'^' h+ e'^'T, 2 < (2.2-5) 



