Water wave propagation through arrays of closely spaced surface-piercing vertical barriers

Abstract

This paper presents and compares two different approaches to solving the problem of wave propagation across a large finite periodic array of surface-piercing vertical barriers. Both approaches are formulated in terms of a pair of integral equations, one exact and based on a spacing $\delta > 0$ between adjacent barriers and the other approximate and based on a continuum model formally developed by using homogenisation methods for small $\delta$ . It is shown that the approximate method is simpler to evaluate than the exact method which requires eigenvalues and eigenmodes related to propagation in an equivalent infinite periodic array of barriers. In both methods, the numerical effort required to solve problems is independent of the size of the array. The comparison between the two methods allows us to draw important conclusions about the validity of homogenisation models of plate array metamaterial devices. The practical interest in this problem stems from the result that for an array of barriers there exists a critical value of radian frequency, $\omega _c$ , dependent on $\delta$ , below which waves propagate through the array and above which it results in wave decay. When $\delta \to 0$ , the critical frequency is given by $\omega _c = \sqrt {g/d}$ , where $d$ is the plate submergence and $g$ is the acceleration due to gravity, which relates to the resonance in narrow channels and is an example of local resonance, studied extensively in metamaterials. The results have implications for proposed schemes to harness energy from ocean waves and other problems related to rainbow trapping and rainbow reflection.