Modulational instability of a pair of collinear wave trains

A coupled system of three nonlinear evolution equations is derived for a pair of collinear wave packets over finite depth fluid. The wave packets are narrow banded having different carrier wave frequencies. The evolution equations are employed to perform stability analysis of a pair of collinear wave trains. It is observed that the region of instability as well as the growth rate of instability for counter-propagating waves is greater than that for co-propagating waves when everything else remains same. The region of instability increases with the increase in the depth of the medium. It is also found that the growth rate of instability of a wave train increases with the increase in wave-steepness of the second wave train.