CIRC ULA R C I UNDER CA VIIY RESONA TOR 37 



Since both streamlines and contours are periodic in z and 6, it is not 

 essential to represent more than is covered in a rectangular piece of the side 

 wall corresponding to quarter periods in :: and d. These are covered in a 



L . ttD 



length T~ along the cavity and in a distance ~t~ around the cavitv. If 

 2h 4' 



such a piece of the surface be rolled out onto a plane it forms a rectangle 



irnD 

 of proportions ~. . 



The ditliculty in depicting the side wall currents of TE modes, as com- 

 pared with the end plate currents, is now apparent. For the end plate, the 

 "proportions" are fixed as being a circle. Furthermore, for a given f, as 

 m increases the effect is merely to add on additional rings to the previous 

 streamline and contour plots. Here, however, the proportions of the rec- 

 tangle are variable, in the first place. And for a given rectangle the stream- 

 lines and contours both change as ( and )n are varied. Another way of ex- 

 pressing the same idea is that for end plates the current distribution does 

 not depend upon the mode index n, and varies only in an additive way with 

 the index m, whereas for the side walls the distribution depends in nearly 

 equal strength on f, m and ;/. 



Some simplification of the situation is accomplished by introducing two 

 new parameters, the "shape" and the "mode" parameters, defined by: 



irnD ( 



S = — M=^ (17) 



and two new variables 



Z = hz <f> = (d. (18) 



Substitution of the above, and also the expressions for k\ and ^3 (see Fig. 

 1) into (15) and (16) yields 



cos Z = C(cos (/)) (streamlines) (19) 



T-2 2 . ni/2 



cos Z 



{S^M^ sin2 4> - cos- <^). 



(contours). (20) 



For given proportions S, one can calculate the streamlines and contours for 

 various values of M. Thus a "square array" of side wall currents can be 

 prepared, such as shown on Fig. 2. 



The mode parameter, if, in the physical case takes on discrete values 

 which depend on the mode. Some of its values are given in the following 

 table. They all lie between and 1 and there are an infinite number of 

 them. 



