322 BELL SYSTEM TECHNICAL JOURNAL 



In these equations the infinite square matrix tanh cV is defined as (sinh cT) 

 (cosh cT)~^ and coth cT as its reciprocal, sinh cT and cosh cT may be de- 

 fined as power series in cT and may be expressed as combinations of exp (cT) 

 and exp (—cT). 



An expression for the column matrix /~ may be obtained by adding the 

 equations in (2.3-2), Before doing this it is convenient to introduce the 

 two column matrices x and y defined by 



{VcT tanh cT + cTq) x = cFo e^' h 



(2 ?>-?)) 

 {VcV coth cV -f cFo) y = cTo e^' h 



where the scalar length c has been introduced to make the various terms 

 dimensionless. Each equation in (2.3-3) represents an infinite set of 

 simultaneous linear equations to be solved for the elements of x or y. 

 Once X and y are known the reflected wave is given by 



(2.3-4) 



(2.3-5) 



PART III 

 GENTLE BENDS— GENERAL THEORY 



3.1 Limiting Forms Assumed for T and V 



It will be shown in Part IV that for gentle circular bends in rectangular 

 wave guides the matrix propagation constant T is such that 



r' = tI + F (3.1-1) 



where To is the square of the matrix propagation constant for the straight 

 guide. To is a diagonal matrix having 8m (which is one of the r„,„'s, depend- 

 ing on the set of modes under consideration, given by (1.1-5)) for the w*^ 

 element in its principal diagonal. F is a square matrix of infinite order in 

 which the elements Fa in the principal diagonal are of order ^^ and the re- 

 maining elements Fij, i 9^ j are of order ^. Here ^ = a/pi is the ratio of the 

 guide width to the radius of curvature of the bend. As the bend becomes 

 more and more gentle, ^ -^ 0. 



The asymptotic expressions given in Appendix I show that, for gentle 

 bends in rectangular guides, the square matrix V which appears in the 

 junction conditions (2.2-7) approaches a unit matrix as ^ -^ 0. In par- 

 ticular Vii = 1 -f Vii where va is of order ^'^, and Va, the element in the i*^ 

 row andy* column, is of order ^ when i 9^ j. 



