  Bloch wavefunctions are used to derive dispersion relations for water wave
propagation in the presence of an infinite array of periodically arranged
surface scatterers. For one dimensional periodicity (stripes), band gaps for
wavevectors in the direction of periodicity are found corresponding to multiple
Bragg scattering. The dependence of these band gaps as a function of scatterer
density, strength, and water depth is analyzed. We find in contrast to band gap
behavior in electronic, photonic, and acoustic systems, these gaps can increase
with excitation frequency $\omega$. Thus, higher order Bragg scattering can
play a dominant role in suppressing wave propagation. Furthermore, in one
dimension, an additional constraint (in addition to single scatterer energy and
momentum conservation) on the calculation of transmission $T$ and reflection
$R$ coefficients of a finite number of scatterers is calculated from the exact
dispersion relation of the infinite periodic system. This relationship may be
useful in measurements where only phase or amplitude is easily measured. In
simple two dimensional periodic geometries, we find no complete band gaps and
offer semi-quantitative reasons for why this is so. The role of evanescent
modes is discussed, and finally, equations for water wave band structure in the
presence of a uniform flow are derived.
