Bragg scattering of water waves by a doubly periodic seabed

We extend the recent work on Bragg scattering of water waves by one-dimensional parallel bars of sinusoidal profile to two-dimensional, doubly sinusoidal bed waves. The resonance condition governing the phase matching between the incident, scattered and bed waves is now more complicated and a much richer variety of resonant reflection can occur. In particular, for one normally incident wave there can be two reflected waves forming a standing wave in the transverse direction. Solutions for a wide strip of bed waves are discussed for incident water waves satisfying approximately the Bragg resonance condition. Modifications for a two-dimensional array of hemispheroids are also given. Possible application to the design of submerged breakwaters is suggested.