GUIDED-WAVK PKOPAGATIOX THKOl (UI (iYROMACiXETIC MFODIA (iOo 



in the propagation theory, as also does a = I. The following scheme 

 exists: for < <x < ao , k^ and fj. are both positive; for (Tu < o" < 1, k < 

 and n < 0, for a > 1, ^ is negati\'e and ji posilixc. if a is changed to 



— (7, /x goes into ju, and k into —k. 



The procednre which will now l)e used to discuss the solution of the 

 characteristic ecfuation, ohserxing the Polder relations, begins by writing 

 the equation, for a, p positi\'e, in the form 



G(Xi , <r, To) = G{T(\i), (X, fo) 



We are already' in possession of a contour map of the left hand side 

 of this eciuation in the quadrant cr > 0, X > 0, and of the function 

 ^(Xo , a, To) in the quadrant X < 0, o- > 0. The latter surface has now to 

 be transformed into one in the Xi-(|uadrant by the relation 



X, = T(Xi) = (a -h p - Xi)/(1 - aXi) 



(or equally well, Xi = T{\2). This may be effected by considering the 

 transformation of curves G(\2 , o-, Vq) = constant, onto the Xi-quadrant. 

 For the /' curves whose analytical expression in terms of a and X2 is very 

 simple, the corresponding explicit expression of the transformed curve 

 in Xi and a is simple. Contours other than /' are most easily transformed 

 l)y replotting G(\2 , c, ro) = const, in the hyperbola-mesh formed by the 

 lines r(X2). However, information about particular points and about 

 asjTiiptotic behavior of these transformed curves is available in analytic 

 form and is stated in Appendix II. The two surfaces so obtained vAW in- 

 tersect in various curves, along Avhose projections on the X — o- plane both 

 Polder relation and (r-equation are satisfied. For each such projection Xi 

 is a function of <x, X2 is then known in terms of a and p, and finally (5" = 



— X1X2 is known. In most cases the general course of these curves can be 

 found without resort to much numerical analysis. Each of the curves is 

 associated with a definite mode and it follows that the classification of 

 the modes can be carried out fairly easily. The approximate location of 

 the solution curves relies upon the fact that if the position of the infinity 

 curves of both surfaces is known, continuity considerations will fre- 

 fiuently assure the existence of an intersection within certain regions. 

 ^Moreover, the neighborhood of certain special points on these solution 

 curves can be investigated analytically. These are points at which one 

 or both of the G-functions may be approximated by a simpler expression; 

 included among these is the point at infinity. 



It is clear that for o and p negative the whole procedure outlined above 

 may be carried out in a similar way, with the o- > 0, X > (luadrant now 

 being transformed on to the cr < 0, X < quadrant. 



