The coupling of the accelerating cavities to dummy cavities was recently found to reduce beam breakup growth. This article analyzes a more sophisticated model that includes the time delay between coupled cavities and covers the possibilities of wave cutoff and resonance in the coupling path. A dispersion relation is obtained. We show that the peak spatial exponentiation rate of the dominant beam breakup mode is reduced by as much as a factor of 2, and this reduction is insensitive to the coupling path length. Other modes are destabilized, however, but they have lower growth rates, in general.