The effects of surface tension on modulational instability in full-dispersion water-wave models

Abstract

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 Hur and Pandey (2016). 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 Hur and Pandey (2016). We also compare the results with the Whitham and full-dispersion Camassa–Holm equations. We conclude that the shallow water model in Hur and Pandey (2016) is a better model than the shallow water model presented here when the effects of surface tension on modulational instability is considered.