HYPER-FREQUENCY TRANSMISSION 333 



Thus in the hmit the roots of (68) will be exactly those of Jn~i{p) = 0. 

 In other words as q varies from to oo the corresponding value of p as 

 given by (68) will not change much. It might appear that the limiting 

 values of p could be roots of Jn^\{p) = 0; this is not possible, however, 

 because in the process of transition p would have to pass th rough the 

 intermediate zero of Jn{p) and no real value of q is consistent with 

 such zeros. 



The case n = I requires a special examination. After multiplying 

 (68) by q- and permitting (/ to approach zero, we find that the first term 

 tends to infinity while the last term becomes a constant. Since the 



limit of ^^ is finite, Ji(p) must approach zero. Thus for w. = 1, the 



critical frequencies are determined by the zeros of Ji{p). 



One interesting point may be mentioned in conclusion. If the guide 

 were surrounded by a hypothetical medium of zero dielectric constant, 

 equation (57) for the £o-waves would become 



/i(Xia) ^ Q^ j^^^^^^ ^ Q (72) 



\iaJo{\ia) 



Thus the critical frequencies would be given by the roots of Jiip) = 



and not by those of Jo{p) = as is the case for any ratio — different 



from zero no matter how small it may be. Our first impression is that 

 this result does violence to our physical common sense which demands 

 that the hypothetical idealized case should be an approximation to the 

 real one when one dielectric constant is large in comparison with the 

 other. And indeed common sense is justified if one does not adhere too 

 closely to the exact mathematical definition of the expression "critical 

 frequency." In the region between any particular zero of Jo(p) , giving 

 the true critical frequency, and the corresponding zero i* of Jiip), 

 giving the "approximate" critical frequency, most of the energy 

 travels outside the guide, with a velocity substantially equal to that of 

 light in the surrounding medium. The "approximate" critical 

 frequency marks the region of the most rapid transition from wave 

 propagation outside the guide to that inside the guide. 

 " This zero is always larger than that of Jo(p)- 



