GENERALIZKD TKLKGItAl'llIS T S EQUATIONS 791 



equations m the following;" form 



= —Z(m)in)I(n) — Z(.m)[n]T[n] — 7\„i) („) T' („) — ?'(»,)[«] l'^[„l , 



dv (m) 



dz 



= — F( „,)(,.) F(„) — Y{m)[n]V[„] — Tim){n)I{n) — 'L\m){n\I [n] , 



d/(„o 



dz 



(30) 



dV[m\ 



dz 



dl[m] 



dz 



— ~Z[m]l7i]I[n] — Z[m]{n)I{n) — T[,„][n]^ [n] — T [,n]{n)V (n) , 



l[m][nlT [«] — y[m]{n)V(„) — T[„,][n]T[n] — T [„,]{„) I (n) • 



Th(> ti'iuisfcr impedances Z, i\\v Iranster admittances F, the voltage 

 transfer coefficients 7', and the current transfer coefficients 'T between 

 \arious modes are in general functions of z. They are constants if the 

 properties of the waveguide are independent of the distance along it; 

 in this case the problem of solving the generalized telegraphist's equa- 

 tions reduces to solving an infinite system of linear algebraic ecjuations 

 and the corresponding characteristic equation. 



Similar ecjuations may be derived for spherical waves either in an un- 

 limited medium or in a medium bounded by a perfectly conducting coni- 

 cal surface of arbitrary cross-section. If the latter is circular and if the 

 flare angle is 180°, we have a plane boundary. Hence, the case of spheri- 

 cal waves in a non-homogeneous medium is included. In the spherical 

 case we shall use the general orthogonal system of coordinates (r, u, v) 

 where r is the distance from the center and {u, v) are orthogonal angular 

 coordinates. In this system the elements of length ds and area dS are 

 given bv 



2/272. 2 J 2^ , , * 2 



ds = dr" -\- r {e\ du + e^ dv"), dS = r dQ, c/O = CiC-^ du dv. {\^\) 



The trans\'erse field components may be expressed in a form similar 

 to that foi- waveguides 



rEt = - grad V - flux n, rHi = flux n - grad U, (32) 



where grad and flux of a typical scalar function are defined l)y ecjuations 

 (10). Instead of (11) we have 



V = -VUr)TUu, v), n = -/(„)(r)r(„)(u, v), 



(33) 

 ^ = -F[„](r)r[„](7A,.), U = -I,„(r)T^.,(n,v), 



where the ^'-functions satisfj^ e(iuation (4) and appropriate boundary 

 conditions. These functions, their gradients and fluxes are orthogonal. 



