Wave scattering by periodic arrays of breakwaters

Oblique incidence of plane waves upon an infinite array of in-line periodic screens or breakwaters in finite water depth is considered using linear water-wave theory. The number of reflected or transmitted waves is a function of the angle of incidence and the ratio of wavelength to array periodicity. A simple matrix formulation is provided for all the reflection and transmission coefficients arising from a particular set of parameters, using a formulation based either on the unknown velocity through a gap or on the unknown pressure difference across a breakwater screen. Integral properties of functions related to these unknowns form the basis of the matrix structure, the functions themselves satisfying a set of integral equations which are solved using a Galerkin approximation that gives highly accurate approximations with very few terms in the expansion. The problem is extended to consider two identical parallel arrays and it is shown that there exists zeros of both reflection and transmission. Finally, a wide-spacing approximation is derived for two arrays based on the accurate results found from the single array problem, where the two arrays do not have to be identical, but must have the same periodicity.