LAMINATED TRANSMISSION LINES. 11 1161 



number of real roots, 



Xi , X2 , X3 , • • • , Xp , • • • , (430) 



of which xi corresponds to the principal mode and X2 , Xs , • • • , to 

 the higher stack modes. The x's are the eigenvalues of the system of 

 equations (369), and as such may be located approximately with a 

 differential analyzer, or as accurately as desii-ed by numerical solution 

 of equation (429). The attenuation and phase constants of the pth 

 mode are 



^' (431) 



2VM/e g 



13 = co\/^ , (432) 



provided that the attenuation per radian is small, i.e., that p is not too 

 large. The fields are given by writing Xp for xi and jp for y in equations 

 (371) to (373) of Section IX. 



For a Clogston 2 with no main dielectric we can set pi = p2 in equation 

 (429) and obtain the much simpler form 



Ji(xa)N,(xh) - Ji(xb)Nr(xa) = 0. (433) 



The pth root of (433) may be written^* 



— a 



where the functions /p (a/6) have values slightly greater than unity when 

 a/b = 0, and decrease monotonically toward 1 as a/6 approaches unity. 

 The attenuation and phase constants of the pth mode in a Clogston 2 

 are given by 



pVflia/b) 

 2V^eg{h-af' ^'^^^ 



"ni , (436) 



provided that p is not too large. The attenuation constant of the pth 

 mode is thus approximately p times the attenuation constant of the 

 principal mode, the approximation being better the closer the ratio 

 a/h is to unity. The fields of the pth mode may be obtained by wTiting 

 Xp for xi and jp for 7 in equations (302) of Section VIII, or equations 

 (311) if a = 0. Qualitatively these fields are very similar to the fields of 



2< Reference 18, pp. 204-206. What we call pirfp(a/h) is tabulated by Jahnke and 

 Emde as (k — l)x/''', where k = b/a. 



