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Although there has been a sustained interest in unstable open resonators for high-power laser applications, no satisfactory theory has previously been developed to explain the intricate behavior of the complex eigenvalues obtained by numerical solution of the resonator integral equation. The present study proposes a remarkably simple model that predicts the basic features of the published results. In this model, a two-dimensional resonator with finite convex mirrors is regarded as an open-ended waveguide with hyperbolic boundaries. Through use of the modal fields in the waveguide region and modal reflection and coupling coefficients due to the open end, one may construct a resonance equation for the eigenvalues of the free resonant fields (eigenmodes) in the unstable resonator. From a study of the resonance equation as a function of mirror size, it may be concluded that a single waveguide mode is primarily responsible for the eigenmode behavior near a loss minimum, and that two selectively coupled waveguide modes provide the behavior in the intervals between loss minima. The selection of the appropriate waveguide modes is accomplished with the aid of a resonance chart for the simpler problem wherein mode coupling at the open waveguide end is omitted. Results are presented and compared with the earlier literature.