  We analyze the Benney model for interaction of short and long waves in
resonant water wave interactions. Our particular interest is in the periodic
traveling waves, which we construct and study in detail. The main results are
that, for all natural values of the parameters, the periodic dnoidal waves are
spectrally stable with respect to perturbations of the same period. For another
natural set of parameters, we construct the snoidal waves, which exhibit
instabilities, in the same setup.
  Our results are the first instability results in this context. On the other
hand, the spectral stability established herein improves significantly upon the
work Angulo-Corcho-Hakkaev, which established stability of the dnoidal waves,
on a subset of parameter space, by relying on the Grillakis-Shatah theory. Our
approach, which turns out to give definite answer for the entire domain of
parameters, relies on the instability index theory. Interestingly, end even
though the linearized operators are explicit, our spectral analysis requires
subtle and detailed analysis of matrix Schr\"odinger operators in the periodic
context, which support some interesting features.
