RECTANGULAR WAVE GUIDES 147 



where Z,„„ = Z„,„ and I',,,,, = I',,,,, is, as in (-^-l^), 



.V 



V,n = T.S,„< \e^' di: + e^' d"[\ 



X 



(=\ 



Here ^c is the integral of ip( as given by (3-20), and ip( is determined by 

 setting the determinant of the matrix v?-/ — ZV to zero. When<|0/ is known, 

 S,n( and T„,f are determined (to within a plus or minus sign which may be 

 absorbed by the constants d( of integration) by the relations 



n=l 

 X 

 (PfTmt = — Z^ ymnSmt (3-34) 



71=1 



.V 



2-j Sm( Tml — 1 



n=l 



The last condition, which arises from the condition that the equations for 

 the second-order terms be consistent, may be regarded as a generalization 

 of Slater's^^ result for the case .Y = 1. 



4. Transjormalionfor Wave Guide Plus Horn 



The system to which we shall apply some of the preceding equations con- 

 sists of a straight wave guide starting at x = — oc and running to .v = 

 where it is connected to a sectoral horn. The horn is flared in the (.v, y) 

 plane only. The dimension of the system normal to the (.v, y) plane is 

 constant and equal to a or i according to whether the electric or magnetic 

 vector is in the plane of the horn. 



One might expect that the field in this system may also (in addition to 

 our method) be determined by an alternating procedure of the type described 

 by Poritsky and Blewett"' using the equations obtained by Barrow and 

 Chu" for transmission in the horn. However, we shall not investigate this 

 j)ossibility as we are primarily interested in using the system as an example 

 to which we may apply the foregoing equations. 



If the total angle of the horn is lair, and if the sides of the straight guide 

 are at y = and y = b, (assuming the electric vector to be in the plane of the 

 horn), the equation of the lower side, i.e., the continuation of the side y = 0, 

 of the horn is y = — .v tan air and that of the upper side is y = ^ + .v tan ar. 

 li z = X -\- iy and w = v -\- id then the Schwarz-Christofifel transformation 



