632 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



We can substitute for dH/dzdt from (1.12) and obtain 

 dz^ dz^dt \ ml dz^df 



a^F aV 



(1.14) 



Thus, we have obtained a linear partial differential equation in F, z and /. 

 So far, nothing has been said about waves or wavelike behavior. We 

 might solve (1.14) for any boundary conditions on V and its derivatives 

 that we chose, by any means, as by using a differential analyzer or a digital 

 computer. There is, however, a well-established technique for dealing with 

 linear partial differential equations with constant coefficients, such as (1.14) 

 is. It is known that they have solutions of the form 



V = Ae^^^'e-i^' (1.15) 



As (1.14) is an entirely real equation, if (1.15) is a solution, the real part 

 of (1.15) is also a solution, i.e.. 



Re {Ae^'^^e-^^') 



is a solution. Hence, we may regard the real part of the complex V as the 

 true physical solution. 

 If we substitute (1.15) into (1.14) we obtain 



ulp* - 2«ocoiS' + ( T - uILC - a-lA oiY 



\^ m / (1.16) 



+ 2woICco'j8 - LCo)' = 



Now (1.16) is an algebraic equation in w and jS. How are we to interpret it? 

 Suppose we are interested in devices driven from sinusoidal generators, 

 such as amplifiers.* This means that co is real, and that it is the radian fre- 

 quency of the applied signal. We may then regard (1.16) as an equation in 

 /3, and, as it is a fourth degree equation, there will in general be four roots. 

 We may regard these as pertaining to four waves, whose voltages vary as 



Fi = Aie'^'''-^''^ 



V2 = A^e'^"''^''^ 



V, = A.e'^'"'-^''^ 



V, = A.e'^'"'-^''^ 



* We might, on the other hand, be interested in devices with an imposed spatial pat- 

 tern, as in a magnetron oscillator. In this case we might assume /3 as a given, real quantity 

 and solve for real or complex values of w. 



