WAVE PICTURE OF MICROWAVE TUBES 1369 



signs while the lower sign applies if the power flows have opposite signs. 

 dq and dp are phase lags per coupling period associated with the two orig- 

 inal modes and ^i and Oo are phase angles associated with the coupling 

 device. 



We can treat the case of continuous coupling by letting the period of 

 coupling L be very short, the angles dq , dp , 6i , 6^ be A-ery small, and the 

 coupling per period, k, be very small. In this case the cosine can be 

 represented by the first terms of a power series and we find that the 

 phase constants ^ of the modes are given by 



^-n^-{'^1V^4^)' (.3) 



Here ^a and ^b are the phase constants for K ^ (zero coupling) 



^. = '-^ (D4) 



(D5) 



L 

 and K is the coupling per unit length 



/^ = I (D6) 



As before, the upper sign in the radical applies when the power flows 

 have the same signs and the lower sign when the power flows have 

 opposite signs. 



In applying (D3) to the case of traveling- wave tubes and backward- 

 wave oscillators, the effect of all but two modes was of course neglected 

 when the two phase constants would have had nearly the same value in 

 the absence of coupling; the curves for such regions were then joined 

 smoothly to give the overall plots of Figs. 11 and 13. 



In Fig. 11 the parameters chosen arbitrarily were: 



/3o = 1 



/3s = u/uo + ^i 



/?/ = 0}/Uo — }i 



K = 0.1 



The complex portion of the phase constant, or, the real portion of the 

 propagation constant, in a stop band caused by the coupling of two 



