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In this paper, we study the instability of highly-oscillating solutions tosemi-linear hyperbolic systems. A instability criterion was given in \cite{Lu}under rather strong separation conditions of resonance sets: coupled resonancesets are pairwise disjoint. Here we show that such separation conditions in\cite{Lu} can be relaxed: one of the coupled non-transparent resonance sets isallowed to intersect with at most two others. We obtain the same instabilitycriterion as in \cite{Lu}. Finally, we give some applications to coupledKlein-Gordon systems with equal masses and nonlinear terms specifiedparticularly, where on the intersections of resonance sets, the relatedinteraction coefficients are non-transparent.

A note on separation conditions of resonance sets in the instability analysis for high-frequency oscillations in geometric optics