314 BELL SYSTEM TECHNICAL JOURNAL 



where 



r!, = p-' {tI + s) (1.3-5) 



It may be verified by direct differentiation of the series (1.1-11) defining 

 exp (— 2:r) that a solution of (1.3-4) is 



a{z) ^e-^-g"- (1.3-6) 



where, as in (1.1-10) for the straight guide, g+ is a column of constants (of 

 integration). However, now Fa is to be obtained by taking the square root 

 of the right hand side of (1.3-5\ a process which is not easy since it usually 

 requires one to obtain the characteristic roots and modal columns of r« 

 (see equation (1.3-10)). 



As far as (1.3-6) being a solution of the differential equation is concerned, 

 Ta may be any matrix whose square is given by (1.3-5). We shall restrict 

 it as follows: When ^ = a/ pi becomes small, as in the case of a gentle bend, 

 it is seen from (1.2-10) and (1.2-11) that P approaches the unit matrix and 

 S approaches zero. Hence, Ta approaches the diagonal matrix Tq. Ta is 

 chosen so that it approaches To, that is, all of the elements in the principal 

 diagonal are either positive real or positive imaginary. This makes 

 exp{—zTa) approach the diagonal matrix exp( — zFo). With this choice 

 of Ta the expression (1.3-6) for a{z) corresponds to a wave traveling in the 

 positive 2 direction. 



The various modes of propagation in the bend may be obtained from T « 

 by expressing, in matrix notation, the steps leading to (1.2-13) (which gives 

 A for they*^ mode). We assume a(z) to be the column matrix obtained by 

 multiplying the column matrix g of constants by the scalar quantity e^^ 

 Setting this in (1.3-4) gives 



{yV - Tl)g = (1.3-7) 



where / is the unit matrix. In order that (1.3-7) may have a solution, the 

 determinant of the coefficient of g must vanish. This leads to the char- 

 acteristic equation* for 7^: 



\y'I-Tl\ = (1.3-8) 



The vertical bars denote the determinant of the inclosed matrix. The roots 

 Ti, 72, • • • are therefore the latent (or characteristic) roots of F^. If we let 

 kj denote** the column g obtained when y = jj in (1.3-7) then 



* See Section 3.6 of Reference'. 



** We choose this notation in order to adhere as closely as possible to that of Refer- 

 ence*. Incidentally, the column kj is proportional to the j^^ column of k"* where k is 

 the modal row matrix introduced in Section 5.1. 



