Continuum modeling of coupled oscillator arrays with coupling delay

The previously published linearized continuum model of a linear (one-dimensional) array of mutually injection locked oscillators is generalized to include the effects of time delay in the interoscillator coupling. This generalization is motivated by a desire for causal solutions describing the dynamics of the phase distribution in the aperture of a phase array antenna driven by such an array of oscillators. The solutions for the phase dynamics will, in general, be noncausal unless the coupling delay is taken into account. In the present formulation the coupling delay is represented by an exponential factor introduced in the Laplace transform of the partial differential equation arising in the continuum model. As a result of this, the transform of the phase distribution exhibits an infinite set of branch points in the complex frequency plane and the inverse transform is computed as the sum of the integrals around the branch cuts.