238 BELL SYSTEM TECHNICAL JOURNAL 



The best procedure seems to be to analyze the situation in a way we know 

 to be vahd, and then to make such approximations as seem reasonable. One 

 approximation we can make is, for instance, that the phase velocity of the 

 wave is quite small compared with the speed of light, so that 



|ri|2»|3o = {o^/cY (6.5) 



In this chapter we shall consider a lossless circuit which supports a group 

 of transverse magnetic modes of wave propagation. The tinned structure of 

 Fig. 4.3 is such a circuit, and so are the circuits of Figs. 4.8 and 4.9 (assum- 

 ing that the fins are so closely spaced that the circuit can be regarded as 

 smooth). It is assumed that waves are excited in such a circuit by a current 

 in the z direction varying with distance as exp {—Tz) and distributed normal 

 to the z direction as a function of x and y,J{x, y). Such a current might 

 arise from the bunching at low signal levels of a broad beam of electrons 

 confined by a strong magnetic field so as not to move appreciably normal 

 to the z direction. 



The structure considered may support transverse electric waves, but these 

 can be ignored because they will not be excited by the impressed current. 



In the absence of an impressed current, any field distribution in the struc- 

 ture can be expressed as the sum of excitations of a number of pairs of nor- 

 mal modes of propagation. For one particular pair of modes, the field dis- 

 tribution normal to the z direction can be expressed in terms of a function 

 Tn{x, y) and the field components will vary in the z direction as exp(±r„2;). 

 Here the + sign gives one mode of the pair and the — sign the other. If 

 r„ is real the mode is passive; the field decays exponentially with distance. 

 If r„ is imaginary the mode is active; the field pattern of the mode propa- 

 gates without loss in the z direction. 



An impressed current which varies in the z direction as exp(— ^^) will 

 excite a field pattern which also varies in the z direction as exp(— F^'), and 

 as some function of .v and y normal to the z direction. We may, if we wish, 

 regard the variation of the field normal to the z direction as made up of a 

 combination of the field patterns of the normal modes of propagation, the 

 patterns specified by the functions 7r„(.f, y). Now, a pattern specified by 

 TTnC^") y) coupled with a variation exp(±rn5;) in the z direction satisfies 

 Maxwell's equations and the boundary conditions imposed by the circuit 

 with no impressed current. If, however, we assume the same variation with 

 .V and y but a variation as exp(— ^^) with z, Maxwell's equations will be 

 satisfied only if there is an impressed current having a distributioii normal 

 to the z direction which also can be ex-j^jressed by the function 7r,j(.v, y). 



Su[)p()sc we add up the various forced modes in such relative strength 

 and i)hasc that the total of tlic imjjresscd currents associated witli them is 

 equal to the actual impressed current. Then, tlie sum of the fields of these 



