The Continuum Model for Linear Arrays
All of the analysis presented so far has treated each oscillator as a discrete device with an injection port and an output port from which a signal emanates having a discrete phase value relative to a phase reference. For this reason, the mathematical model represented has been termed the discrete model. We emphasize that the discrete model encompasses the dynamic behavior of the oscillator array both nonlinear and, if desired, linearized. No new phenomena are added to this range of capability by means of the formulation to be discussed in the present chapter. However, it will be shown that, provided one is willing to linearize, the so-called “continuum model” offers considerable advantage in terms of insight and applicability of familiar mathematical techniques. Although the continuum model is fundamentally approximate primarily because of the linearization, it nevertheless provides intuitive understanding of the behavior of coupled oscillator arrays with small inter-oscillator phase differences, an important special case in terms of practical application. Moreover, it provides a basis for understanding the impact of nonlinearity when the inter-oscillator phase differences increase beyond the limits of accurate linear approximation.
The continuum model in this context was suggested by Pogorzelski et al. . In essence we replace the index identifying the oscillators with a continuous variable such that, when the continuous variable ...