34 BELL SYSTEM TECHNICAL JOURNAL 



The fields in the — .v direction may be obtained by substituting exp(7x) 

 for exp(— 7.v). 



If the sheet is to roll up properly, the points a on the bottom coinciding 

 with the points b on the top, we have 



lirya sin ^ — l-w^^^a cos ^|/ = linr (3.19) 



where n is an integer. 



The solution corresponding most nearly to the wave on a singly-wound 

 helix is that for n = 0. The others lead to a variation of field by // cycles 

 along a circumferential line. These can be combined with the n = solu- 

 tion to give a solution for a developed helix of thin tape, for instance. Or, 

 appropriate combinations of them can represent modes of helices wound of 

 several parallel wires. For instance, we can imagine winding a balanced trans- 

 mission line up helically. One of the modes of propagation will be that in 

 which the current in one wire is 180° out of phase with the current in the 

 other. This can be approximated by a combination of the n = -\-\ and 

 11= —1 solutions. This mode should not be confused with a fast wave, a 

 perturbation of a transverse electromagnetic wave, which can exist around 

 an unshielded helix. 



Usually, we are interested in the slow wave on a singly-wound helix, and 

 in this case we take n = in (3.19), giving 



7 sin i/' — /3o cos ^ = 



(3.20) 



tan yp = /3o/7 



sin i/' = / 2 , ^2x1/2 (3.21) 



(7 + Po) 



Let us evaluate the propagation constant in the axial direction. From Fig. 

 3.8 we see that, in advancing unit distance in the axial direction, we pro- 

 ceed a distance cos \p in the z direction and sin \p in the y direction. Hence, 

 the phase constant jS in the axial direction must be 



/3 = i8o sin i/' + 7 cos ^ (3.23) 



Using (3.18) and (3.19), we obtain 



^ = (/35 + yY' (3.24) 



7 = 0' - 0iy" (3.25) 



These are just relations (3.2, 33). 



