  We study the modulational instability of a shallow water model, with and
without surface tension, which generalizes the Whitham equation to include
bi-directional propagation. Without surface tension, the small amplitude
periodic traveling waves are modulationally unstable if their wave number is
greater than a critical wave number predicting a Benjamin-Feir type instability
and the result qualitatively agrees with the shallow water model in [HP16].
With surface tension, the result qualitatively agrees with the physical problem
except for the large surface tension limit which is accurately predicted by the
shallow water model in [HP16]. We also compare the results with the Whitham and
full-dispersion Camassa-Holm equations.
