306 ISELL SYSTEM TECHNICAL JOURNAL 



The gentle bend formulas were obtained from the matrix equations by the 

 Hmiting process described in Part III. It seems hkely that the matrix 

 method, which is similar to the method used in an earlier paper on trans- 

 mission line equations, may be applied to other wave guide problems. 

 With this thought in mind, the development of Parts II and III has been 

 couched in general terms. 



The matrices used in the present theory are of infinite order since the 

 guide may support an infinite number of modes of propagation. This fact 

 makes it difficult to justify all the steps in our analysis, and we do not at- 

 tempt to do so.* Despite this lack of rigor, I believe that the procedures 

 given here lead to the correct results since they yield, for gentle bends, 

 expressions obtained by Buchholz and Marshak. Moreover, although 

 numerical results tabulated in Part V were obtained by using matrices of 

 only the second and third order, they indicate a rapid convergence as the 

 matrix order is increased. 



PART I 

 PROPAGATION OF WAVES IN GUIDE 



1.1 Propagation in a Straight Wave Guide 



Rather general expressions for the electric and magnetic intensities E and 

 F in a field are (see pp. 127-128 of Reference ) 



E = — ioifiA + - — grad div A — curl B 

 icoe 



(1.1-1) 



H = curl A + - — grad div B — imeB 



tCOfl 



The field is assumed to vary with the time t as e'"', co is the radian fre- 

 quency, n the permeability and e the dielectric constant (for free space 

 fx = 1.257 X 10~^ henries/meter, e = 8.854 X 10"'^ farads/meter). The 

 vector potentials A and B satisfy the wave equations 



V^A = a' A, V'75 = (t'B 



V = Laplacian operator (1-1~2) 



2 2 



a = 0) fie 



In dealing with bends, it is convenient to choose A and B normal to the 

 plane of the bend. In our notation, this plane is always taken to be the x, 

 z plane so that /I and B are parallel to the y axis. The z axis is parallel 



* Similar questions arise in the rigorous treatment of an infinite set of linear 

 equations. A discussion of this subject is given in Chap. TTI of Reference*. 



