ELECTROMAGNETIC FIELD ANALYSIS 495 



(Maxwell's field equations) are partial differential equations. Their solu- 

 tions would normally involve arbitrary functions; but since the tangential 

 electric intensity vanishes at the conducting boundary of the resonator, the 

 solutions assume a much less arbitrary form involving only an infinite set of 

 arbitrary constants. Particular solutions are sought in the form of products 

 of three functions, each depending on only one coordinate. For paral- 

 lelopipedal cavity resonators the various components of electric and mag- 

 netic intensity are assumed in the form X{x) Y(y) Z{z). By substituting in 

 Maxwell's equations it is found — very fortunately indeed — that X, Y, Z 

 may be obtained as solutions of ordinary differential equations. The 

 boundary conditions at the boundaries of the box x = 0,a; y = 0,b;z = ox 

 are easy to satisfy because we have to work with only one of these three 

 functions at a time. 



In general, however, the problem of calculating oscillation modes is by no 

 means simple; but once these modes have been determined, the problem of 

 forced oscillations as well as free oscillations is practically solved. For 

 instance, a small loop inside a resonator is coupled to the various modes and 

 the coupling coefficients can be determined by evaluating the flux linkages. 



Every physical circuit possesses an infinite number of degrees of freedom 

 and circuits with a finite number of degrees of freedom are abstractions. 

 If we take special measures to concentrate magnetic energy as much as 

 possible in a few regions of the medium and electric energy in a few other 

 regions, we shall have a physical network in which a finite number of oscilla- 

 tion modes will be well separated on the frequency scale from all the rest. 

 If we are concerned only with the frequencies comparable to the natural 

 frequencies of this cluster of modes, we can ignore all the higher modes and 

 for our purposes we may regard the network as a finite network. At these 

 frequencies the infinitely small meshes into which we could subdivide the 

 individual "inductors" (regions of magnetic energy concentration) and 

 "capacitors" (regions of electric energy concentration) will oscillate in 

 unison in groups. 



Briefly wt can summarize the above methods of analysis as follows: 

 The medium supporting the electromagnetic field may be regarded as a 

 three-dimensional network of infinitely small meshes in which every mesh is 

 coupled only to the adjacent mesh. Circuit equations applied to this 

 network lead to Maxwell's differential equations. In contrast with this 

 ^^ natural network model of the medium^' we can construct a reduced network 

 model in which only the conductors of the medium are subdivided into 

 meshes. The medium surrounding the conductors is concealed in the 

 mutual impedances of the constituent meshe3. Every mesh is coupled to 

 every other mesh and the mutual impedance (or the coupling factor) is 



