Multiple resonant scattering of water waves by a two-dimensional array of vertical cylinders: Linear aspects

We study the Bragg resonance of surface water waves by a two-dimensional array of vertical cylinders covering a large area of the sea. Starting from the resonance criterion known in the physics of solid state and crystallography, we employ asymptotic techniques to derive two-dimensional coupled-mode equations for the envelopes of scattered waves resonated by a plane incident wave. Explicit analytical solutions are obtained for a long strip of cylinder array which may be used for supporting a future offshore airport. Examples of both two-wave and three-wave resonances are discussed in detail. Roles of the band gaps are examined.