Water wave propagation and scattering over topographical bottoms

Here we provide a derivation of the formula recently used for investigating Bragg resonance in waves on a shallow fluid over a periodically drilled bottom [M. Torres et al., Phys. Rev. E $63,$ 011204 (2000)]. The equation is also compared with other existing theories. As an application, the theory is extended to the case of water waves propagating over a column with an arbitrary array of cylindrical steps. For a regular array, the formulation for computing band structures is also presented.