Water-wave diffractions by three-dimensional periodic rectangular blocks: Fourier modal analysis

Abstract

Water-wave diffractions caused by three-dimensional (3D) periodic rectangular blocks are analytically modeled with a Fourier modal analysis method using full-linear potential theory. The boundary conditions in the dimension of periodic extent are conjugated into the other two dimensions in terms of Toeplitz matrices. The elements of the Toeplitz matrices are the Fourier harmonics of vertical and horizontal eigenfunctions converted to special-frequency domains. This allows us to reduce the 3D model system to an ordinary two-dimensional counterpart. The boundary matchings finally lead to a general form of block centrosymmetric matrix for the rectangular topography. A fair consistency between the present results and the data from the literature validates such a Fourier modal analysis. Effects of different parameters on multiple-modal diffraction efficiencies of reflected and transmitted wave energy are computed and analyzed. This study is useful for the designers of coastal breakwaters and focusing lens of water waves.