Viscous effects on Bragg scattering of water waves by an array of piles

We examine the effects of viscous damping in the boundary layers on the Bragg resonance of surface water waves by a two-dimensional array of vertical cylinders. For cylinders of small radius relative to the wavelength, we first derive an effective boundary condition for the radial derivative of the velocity potential to account for the viscous forces. Coupled-mode equations are then rederived by an asymptotic method for the envelopes of multiply resonated waves inside the array. Effects of viscosity on band gaps and scattering coefficients due to a plane incident wave are examined analytically for an infinitely long array of finite width surrounded by open water. For normal incidence the envelope physics is one dimensional. The transmission and reflection properties are studied first. Oblique incidence can in principle excite several wave trains in different directions. Explicit solutions are given and discussed when there are only two wave trains inside the array. Results are compared with recent theories where viscosity is not taken into account. The asymptotic theory can be modified for two-dimensional sound scattering by a cylinder array.