Band structure of surface flexural–gravity waves along periodic interfaces

We extend Floquet's Theorem, similar to that used in calculating electronic and optical band gaps in solid state physics (Bloch's Theorem), to derive dispersion relations for small-amplitude water wave propagation in the presence of an infinite array of periodically arranged surface scatterers. For one-dimensional periodicity (stripes), we find band gaps for wavevectors in the direction of periodicity corresponding to frequency ranges which support only non-propagating standing waves, as a consequence of multiple Bragg scattering. The dependence of these gaps on scatterer strength, density, and water depth is analysed. In contrast to band gap behaviour in electronic, photonic, and acoustic systems, we find that the gaps here can increase with excitation frequency ω. Thus, higher-order Bragg scattering can play an important role in suppressing wave propagation. In simple two-dimensional periodic geometries no complete band gaps are found, implying that there are always certain directions which support propagating waves. Evanescent modes offer one qualitative reason for this finding.