Scattering of water waves by multiple rows of vertical thin barriers

The reflection and transmission of waves by a periodic array of thin fixed identical vertical barriers extending uniformly through the depth of the fluid is considered. The water wave problem, which also has analogues in acoustics and electromagnetics, involves scattering by a large finite number of equally spaced rows, each consisting of an array of barriers with a linear periodic arrangement extending to infinity. The main purpose of the work is to compare the results of two approximate methods of solution based on homogenisation techniques with exact results allowing us to determine when the results can be expected to be in good agreement. When no approximation is made, a method of solution (Discrete Model) describing the propagation of incident waves through the exact barrier arrangement is derived using Floquet-Bloch eigen solutions for the corresponding infinite periodic array. It is shown how the solution can be reduced to a simple pair of integral equations which results only from matching the region containing barrier arrays to the two exterior domains. Numerical solutions are, nevertheless, relatively complicated and two approximate models are developed which considerably simplify the numerical effort required. In one approximation (Continuum Model I) it is assumed only that the spacing between two adjacent rows of barriers is small and is complementary to the commonly-used wide-spacing approximation. The complication of the geometry is removed by replacing the governing equations and boundary conditions with an effective continuous medium and a simplified pair of integral equations now govern the solutions. In a second approximation (Continuum Model II), a low-frequency homogenisation approach based on a long-wave assumption is invoked and matching now leads to closed-form expressions for the reflection and transmission coefficients. Results concentrate on the comparison between different methods which allows us to assess the validity of each continuum model. This understanding will be important in developing approximate models that describe the operation of wave energy farms consisting of large arrays of oscillating flaps designed to absorb energy.