Bragg reflection of infragravity waves by sandbars

Using a multiple-scale perturbation method we derive a set of governing equations describing the transformation of long wave and short wave components in a wave group. These equations are derived from Boussinesq equations with the assumption that the length scale of wave group modulation is in the same order of magnitude as that of the bottom variation, and is much longer than the length scale for the carrier (short) waves. Therefore the reflection of carrier waves by the topographical variation is small and neglected. Numerical examples are presented to show that long waves associated with wave group can be reflected resonantly by a field of periodic sandbars.