RECTANGULAR WAVE GUIDES 137 



method the propagation function g(v, 6) is derived graphically from the 

 geometry of the wave guide irregularities and the result used in one or the 

 other of two sets of differential equations which are equivalent to those 

 derived below. Piloty's work is scheduled to appear soon in the Zeihclirifl 

 fiir angrii'audle Physik under the title "Ausbreitung el.-magn. W'ellen in 

 inhomogcnen Rechteckrohren." 



1. Differential Equations when Electric Vector is in {x, y) Plane 



The partial differential equation to be solved is, from equation (2-3) of 

 the companion j)aper', 



g + + u + sO, «)i*"e = (1-1) 



where 



— = at = and = tt 

 dd 



00 



1 + g{v,e) = 1 + E ancosne = \f'{-v + id) fir'/b" (1-2) 



k — [(26/Xo)" — {b/aYY , Xo = free space wavelength 



In (1-2), z = X -]- iy ^ f{v + id) is the transformation which carries the 

 wave guide system in the (x, y) plane into the straight guide of width 6 = v 

 in the (v, 6) plane. For the sake of simplicity we shall always assume that 

 far to the left the system becomes a straight wave guide of dimensions 

 a, b {b < a) such that only the dominant mode is propagated without 

 attenuation. This insures that the a„'s (which are functions of v) will 

 approach zero a.sv—^ — oo . The dimension (of our system) normal to the 

 {x, y) plane is a throughout. 



Since the normal derivative of Q vanishes on the walls at = and 6 = t 

 we assume 



Q= Fq + FiCosO -\- FiC05 2d+ ••' , (1-3) 



where Fi , F^ , ■ ■ ■ are functions of v, and substitute it together with the 

 Fourier series (1-2) for 1 + g(v, 6) in (1-1). 



The equations obtained by setting the coefficients of the resulting cosine 

 series to zero are 



Fo + (1 + ao)k'Fo + ^ Z <7„/'„ - (1-4) 



2 n=l 



F'rl + [(1 + ao + a2m/2)k' - m']F,n + a^k'Fo (1-5) 



,2 00 



+ ^ Z^' {a\n-m\ + an+m)Fn = 

 Z n=l 



