CURVED WAVE CI IDES 5 



As shown in the analytical part, the couphng coefficient of these slant 

 waves may be defined as the energy interchanged between the modes per 

 unit length of line divided by the geometric mean of the energies per unit 

 length stored up in each of the modes. 



b'rom the coupling coefficient of the slant waves the coupling coefficient 

 of the wave guide modes is derived. 



On the basis of the above physical interpretation the analysis is carried 

 out and the properties of TEoi propagation through curved wave guides of 

 various shapes are derived in the analytical part of this paper which is 

 subdivided into the following nine sections: 



Section 1 develops an approximate theory of loosely coupled, weakly 

 damped circuits. The theory is first derived for coupled resonators which 

 are familiar to communication engineers, and then applied in similar form 

 to coupled transmission lines. It is shown that the important interaction 

 properties of coupled lines are functions of a single coupling discriminant. 

 The relative energy content of the two lines in each of the two possible 

 coupled modes is plotted as a function of the coupling discriminant. 



Section 2 contains the field equations of a straight circular wave guide 

 and their modification by a toroidal bend. 



Section 3 gives the solutions of the field equations for the uncoupled 

 TEoi and TjMu modes in wave guides with infinite, and with small but 

 finite conductivity. 



Section 4 applies the coupling theory to the TEoi and TMh modes in 

 circular wave guide bends. The coupling coefficient, coupling discriminant 

 and energy division between the two modes are deri\'ed as functions of the 

 wave guide diameter bending radius and conductivity and of the signal 

 frequency. 



Section 5 derives the critical bending radius and the attenuation of TEoi 

 waves in long wave guides of constant curvature. Two numerical examples 

 are given. 



Section 6 shows that in a curved section of wave guide which follows a 

 long straight section or other source of pure TEoi the energy fluctuates 

 back and forth between a condition of pure TEoi and of predominant TMu . 

 'llie length and magnitude of the fluctuations are derived. 



Section 7 computes the increase in average attenuation caused by serpen- 

 tine bends of regular shapes. Numerical examples are tabulated. 



Section 8 shows that the results of Section 7 can be applied to helical 

 bends and to small two-dimensional random deviations from a straight 

 course. 



Section 9 shows that for any given statistical distribution of random 

 angular deviations the average attenuation is minimized by an optimum 



