An eigenvalue method based on the integral equation with a Rankine source for the band structure of water surface waves over periodic scatterers

Abstract

An eigenvalue model is developed to obtain the band structure of water surface waves in the presence of an infinite array of periodically arranged scatterers, which include both underwater terrains and fixed bodies on the free surface or in the water. Combined with Bloch's theorem, the scattering of waves over arbitrary terrain or bodies is described as a boundary value problem of the Laplace equation, from which the boundary integral equation can be derived and transformed into an eigenvalue problem through the boundary element method to obtain the band structure of water surface waves. Comparisons are made with various existing models and reflection coefficients from experiments, which shows the validity and accuracy of the proposed model.As an application of the present model, the stopbands for convex parabolic terrain, concave parabolic terrain, sinusoidal terrain and cylinders with centres on the still water surface are calculated. The dependence of the stopbands on the relative size of scatterers and relative water depth is investigated and analysed.