HYPER-FREQUENCY TRANSMISSION 331 



In addition to either of these equations, we have 



7 = VXi2 - MiticoV^' = VX2' - M2e2coVc2, (59) 



and the condition that for truly guided waves y and X2 must be pure 

 imaginary while Xi is real. When X2 is pure imaginary, the Hankel 

 function of the second kind will decrease almost exponentially with 

 increasing distance from the guide if this distance is sufficiently large. 

 If Xi and X2 are taken from (59) and substituted in (57) and (58) we 

 shall have equations determining 7 in terms of w. Unfortunately these 

 equations do not admit of an explicit solution for 7. It is possible, 

 however, to carry out the numerical calculations in the following 

 manner. We plot the left and the right terms of (57), let us say, 

 against their arguments; then we select a pair of values of these 

 arguments corresponding to equal ordinates. Let us suppose that we 

 obtain 



(Xia)2 = p\ (X2o)2 = - q\ (60) 



where p and g are real. Referring to section III, we have p = y and 

 iq = X. Substituting these in (59) and solving, we have 



Hp'^ + q^ _ • /j"2e2Co2 q 



i2 



7 = .^/-^+^,- (61) 



a yj file I — ixi€i 



Since mi usually equals ^2, the guided waves are possible only if the 

 dielectric constant of the guide is higher than that of the surrounding 

 medium. 



The lowest value of q is zero; the right member of (57) is then infinite 

 and the corresponding value of p must then be a root of 



J,{pm) = 0. (62) 



Corresponding to each root w^e have a different mode of propagation. 

 The lowest frequency which can be transmitted in any particular mode 

 and the corresponding propagation constant are given by 



- - ^^- y ^^jd^. (63) 



a^llltl — 1X262 



At this frequency the phase velocity of propagation is equal to that of 

 light in air. Since X2 is small, the field extends to great distances 

 outside the guide. As q increases indefinitely, the right part of (57) 

 approaches zero and p must approach the root of /i(.v) near the par- 

 ticular root of /o that we happen to be considering. Thus for large 

 values of q, we have approximately 



cq i'coV/UiCi /z,.N 



CO = - — = , 7 = i^-ij 



a^Mlel — M2«2 ^ 



